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In [[mathematics]], many '''[[logarithm]]ic identities''' exist. The following is a compilation of the notable of these, many of which are used for computational purposes.

== Trivial identities ==

''Trivial'' mathematical identities are relatively simple (for an experienced mathematician), though not necessarily unimportant. The trivial logarithmic identities are as follows:
{| cellpadding=3
| if <math>\log_b(1) = 0 </math>|| then || <math> b^0 = 1</math>
|-
| if <math>\log_b(b) = 1 </math>|| then || <math> b^1 = b</math>
|}

=== Explanations ===
By definition, we know that:

<math display="block">\log_b(y) = x \iff b^x = y,</math>

where <math>b \neq 0 </math> and <math>b \neq 1</math>.

<!-- PROPERTY 1 -->
Setting <math>x = 0</math>,
we can see that:

<math display="block">
b^x = y
\iff   b^{(0)} = y
\iff   1 = y
\iff   y = 1
</math>

So, substituting these values into the formula, we see that:

<math display="block">
\log_b (y) = x
\iff \log_b (1) = 0,
</math>

which gets us the first property.

<!-- PROPERTY 2 -->
Setting <math>x = 1</math>, we can see that:

<math display="block">
b^x = y
\iff   b^{(1)} = y
\iff   b = y
\iff   y = b
</math>

So, substituting these values into the formula, we see that:

<math display="block">
\log_b (y) = x
\iff \log_b (b) = 1,
</math>

which gets us the second property.

== Cancelling exponentials ==
Logarithms and [[Exponential function|exponentials]] with the same base cancel each other. This is true because logarithms and exponentials are inverse operations{{snd}}much like the same way multiplication and division are inverse operations, and addition and subtraction are inverse operations:<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Logarithm|url=https://mathworld.wolfram.com/Logarithm.html|access-date=2020-08-29|website=mathworld.wolfram.com|language=en}}</ref>

<math display="block">b^{\log_b(x)} = x\text{ because }\mbox{antilog}_b(\log_b(x)) = x</math>
<math display="block">\log_b(b^x) = x\text{ because }\log_b(\mbox{antilog}_b(x)) = x</math>

Both of the above are derived from the following two equations that define a logarithm: (note that in this explanation, the variables of <math>x</math> and <math>x</math> may not be referring to the same number)

<math display="block">\log_b (y) = x  \iff  b^x = y</math>

Looking at the equation <math> b^x = y </math>, and substituting the value for <math>x</math> of
<math> \log_b (y) = x </math>, we get the following equation:

<math display="block">
b^x = y
\iff b^{\log _b(y)} = y
\iff b^{\log_b (y)} = y,
</math>

which gets us the first equation.

Another more rough way to think about it is that <math> b^{\text{something}} = y</math>,
and that that "<math>\text{something}</math>" is <math> \log_b (y) </math>.

Looking at the equation <math> \log_b (y) = x</math>, and substituting the value for <math> y </math> of <math>b^x = y</math>, we get the following equation:

<math display="block">
\log_b (y) = x
\iff \log_b(b^x) = x
\iff \log_b(b^x) = x,
</math>

which gets us the second equation.

Another more rough way to think about it is that <math> \log_b (\text{something}) = x</math>, and that that something "<math>\text{something}</math>" is <math> b^x</math>.
<!--   previously what was there:
<math display="block">b^c = x \iff \log_b(x) = c</math>

Substituting {{mvar|c}} in the left equation gives {{math|1=''b''<sup>log<sub>''b''</sub>(''x'')</sup> = ''x''}}, and substituting {{mvar|x}} in the right gives {{math|1=log<sub>''b''</sub>(''b''<sup>''c''</sup>) = ''c''}}. Finally, replace {{mvar|c}} with {{mvar|x}}.
-->

== Using simpler operations ==

Logarithms can be used to make calculations easier.  For example, two numbers can be multiplied just by using a logarithm table and adding. These are often known as logarithmic properties, which are documented in the table below.<ref>{{Cite web|title=4.3 - Properties of Logarithms|url=https://people.richland.edu/james/lecture/m116/logs/properties.html|access-date=2020-08-29|website=people.richland.edu}}</ref> The first three operations below assume that {{math|1=''x'' = ''b''<sup>''c''</sup>}} and/or {{math|1=''y'' = ''b''<sup>''d''</sup>}}, so that {{math|1=log<sub>''b''</sub>(''x'') = ''c''}} and {{math|1=log<sub>''b''</sub>(''y'') = ''d''}}. Derivations also use the log definitions {{math|1=''x'' = ''b''<sup>log<sub>''b''</sub>(''x'')</sup>}} and {{math|1=''x'' = log<sub>''b''</sub>(''b''<sup>''x''</sup>)}}.

{| cellpadding=3
| <math>\log_b(xy)=\log_b(x)+\log_b(y)</math> || because || <math>b^c b^d=b^{c+d}</math>
|-
| <math>\log_b(\tfrac{x}{y})=\log_b(x)-\log_b(y)</math> || because || <math>\tfrac{b^c}{b^d}=b^{c-d}</math>
|-
| <math>\log_b(x^d)=d\log_b(x)</math> || because || <math>(b^c)^d=b^{cd}</math>
|-
| <math>\log_b\left(\sqrt[y]{x}\right)=\frac{\log_b(x)}{y}</math> || because || <math>\sqrt[y]{x}=x^{1/y}</math>
|-
| <math>x^{\log_b(y)}=y^{\log_b(x)}</math> || because || <math>x^{\log_b(y)}=b^{\log_b(x)\log_b(y)}=(b^{\log_b(y)})^{\log_b(x)}=y^{\log_b(x)}</math>
|-
| <math>c\log_b(x)+d\log_b(y)=\log_b(x^c y^d)</math> || because || <math>\log_b(x^c y^d)=\log_b(x^c)+\log_b(y^d)</math>
|}

Where <math>b</math>, <math>x</math>, and <math>y</math> are [[positive real numbers]] and <math>b \ne 1</math>, and <math>c</math> and <math>d</math> are real numbers.

The laws result from canceling exponentials and the appropriate law of indices. Starting with the first law:

<math display="block">xy = b^{\log_b(x)} b^{\log_b(y)} = b^{\log_b(x) + \log_b(y)} \Rightarrow \log_b(xy) = \log_b(b^{\log_b(x) + \log_b(y)}) = \log_b(x) + \log_b(y)</math>

The law for powers exploits another of the laws of indices:

<math display="block">x^y = (b^{\log_b(x)})^y = b^{y \log_b(x)} \Rightarrow \log_b(x^y) = y \log_b(x)</math>

The law relating to quotients then follows:

<math display="block">\log_b \bigg(\frac{x}{y}\bigg) = \log_b(x y^{-1}) = \log_b(x) + \log_b(y^{-1}) = \log_b(x) - \log_b(y)</math>
<math display="block">\log_b \bigg(\frac{1}{y}\bigg) = \log_b(y^{-1}) = - \log_b(y)</math>

Similarly, the root law is derived by rewriting the root as a reciprocal power:

<math display="block">\log_b(\sqrt[y]x) = \log_b(x^{\frac{1}{y}}) = \frac{1}{y}\log_b(x)</math>

=== Derivations of product, quotient, and power rules ===

These are the three main logarithm laws, rules, or principles,<ref>
{{Cite web
|url=https://courseware.cemc.uwaterloo.ca/8/assignments/210/6
|title=Properties and Laws of Logarithms
|website=courseware.cemc.uwaterloo.ca/8
|access-date=2022-04-23
}}
</ref> from which the other properties listed above can be proven. Each of these logarithm properties correspond to their respective exponent law, and their derivations and proofs will hinge on those facts. There are multiple ways to derive or prove each logarithm law{{snd}}this is just one possible method.

==== Logarithm of a product ====

To state the ''logarithm of a product'' law formally:

<math display="block">\forall b \in \mathbb{R}_+, b \neq 1, \forall x, y, \in \mathbb{R}_+, \log_b(xy) = \log_b(x) + \log_b(y)</math>

Derivation:

Let <math>b \in \mathbb{R}_+</math>, where <math>b \neq 1</math>,
and let <math>x, y \in \mathbb{R}_+</math>. We want to relate the expressions <math>\log_b(x)</math> and <math>\log_b(y)</math>.  This can be done more easily by rewriting in terms of exponentials, whose properties we already know. Additionally, since we are going to refer to <math>\log_b(x)</math> and <math>\log_b(y)</math> quite often, we will give them some variable names to make working with them easier: Let <math>m = \log_b(x)</math>, and let <math>n = \log_b(y)</math>.

Rewriting these as exponentials, we see that

<math display="block">\begin{align}
m &= \log_b(x) \iff b^m = x, \\
n &= \log_b(y) \iff b^n = y.
\end{align}</math>

From here, we can relate <math>b^m</math> (i.e. <math>x</math>) and <math>b^n</math> (i.e. <math>y</math>) using exponent laws as

<math display="block">xy = (b^m)(b^n) = b^m \cdot b^n = b^{m + n}</math>

To recover the logarithms, we apply <math>\log_b</math> to both sides of the equality.

<math display="block">\log_b(xy) = \log_b(b^{m + n})</math>

The right side may be simplified using one of the logarithm properties from before: we know that <math>\log_b(b^{m + n}) = m + n</math>, giving

<math display="block">\log_b(xy) = m + n</math>

We now resubstitute the values for <math>m</math> and <math>n</math> into our equation, so our final expression is only in terms of <math>x</math>, <math>y</math>, and <math>b</math>.

<math display="block">\log_b(xy) = \log_b(x) + \log_b(y)</math>

This completes the derivation.

==== Logarithm of a quotient ====

To state the ''logarithm of a quotient'' law formally:

<math display="block">\forall b \in \mathbb{R}_+, b \neq 1, \forall x, y, \in \mathbb{R}_+, \log_b \left( \frac{x}{y} \right) = \log_b(x) - \log_b(y)</math>

Derivation:

Let <math>b \in \mathbb{R}_+</math>, where <math>b \neq 1</math>,
and let <math>x, y \in \mathbb{R}_+</math>.

We want to relate the expressions <math>\log_b(x)</math> and <math>\log_b(y)</math>. This can be done more easily by rewriting in terms of exponentials, whose properties we already know. Additionally, since we are going to refer to <math>\log_b(x)</math> and <math>\log_b(y)</math> quite often, we will give them some variable names to make working with them easier: Let <math>m = \log_b(x)</math>, and let <math>n = \log_b(y)</math>.

Rewriting these as exponentials, we see that:

<math display="block">\begin{align}
m &= \log_b(x) \iff b^m = x, \\
n &= \log_b(y) \iff b^n = y.
\end{align}</math>

From here, we can relate <math>b^m</math> (i.e. <math>x</math>) and <math>b^n</math> (i.e. <math>y</math>) using exponent laws as

<math display="block">\frac{x}{y} = \frac{(b^m)}{(b^n)} = \frac{b^m}{b^n} = b^{m - n}</math>

To recover the logarithms, we apply <math>\log_b</math> to both sides of the equality.

<math display="block">\log_b \left( \frac{x}{y} \right) = \log_b \left( b^{m -n} \right)</math>

The right side may be simplified using one of the logarithm properties from before: we know that <math>\log_b(b^{m - n}) = m - n</math>, giving

<math display="block">\log_b \left( \frac{x}{y} \right) = m -n</math>

We now resubstitute the values for <math>m</math> and <math>n</math> into our equation, so our final expression is only in terms of <math>x</math>, <math>y</math>, and <math>b</math>.

<math display="block">\log_b \left( \frac{x}{y} \right) = \log_b(x) - \log_b(y)</math>

This completes the derivation.

==== Logarithm of a power ====

To state the ''logarithm of a power'' law formally:

<math display="block">\forall b \in \mathbb{R}_+, b \neq 1, \forall x \in \mathbb{R}_+, \forall r \in \mathbb{R},  \log_b(x^r) = r\log_b(x)</math>

Derivation:

Let <math>b \in \mathbb{R}_+</math>, where <math>b \neq 1</math>, let <math>x\in \mathbb{R}_+</math>, and let <math>r \in \mathbb{R}</math>. For this derivation, we want to simplify the expression <math>\log_b(x^r)</math>. To do this, we begin with the simpler expression <math>\log_b(x)</math>. Since we will be using <math>\log_b(x)</math> often, we will define it as a new variable: Let <math>m = \log_b(x)</math>.

To more easily manipulate the expression, we rewrite it as an exponential. By definition, <math>m = \log_b(x)  \iff  b^m = x</math>, so we have

<math display="block">b^m = x</math>

Similar to the derivations above, we take advantage of another exponent law. In order to have <math>x^r</math> in our final expression, we raise both sides of the equality to the power of <math>r</math>:

<math display="block">
\begin{align}
(b^m)^r &= (x)^r \\
b^{mr} &= x^r
\end{align}
</math>

where we used the exponent law <math>(b^m)^r = b^{mr}</math>.

To recover the logarithms, we apply <math>\log_b</math> to both sides of the equality.

<math display="block">\log_b(b^{mr}) = \log_b(x^r)</math>

The left side of the equality can be simplified using a logarithm law, which states that <math>\log_b(b^{mr}) = mr</math>.

<math display="block">mr = \log_b(x^r)</math>

Substituting in the original value for <math>m</math>, rearranging, and simplifying gives

<math display="block">
\begin{align}
\left( \log_b(x) \right)r &= \log_b(x^r) \\
r\log_b(x) &= \log_b(x^r) \\
\log_b(x^r) &= r\log_b(x)
\end{align}
</math>

This completes the derivation.

== Changing the base ==

Most calculators have buttons for natural logarithms ({{fake link|ln}}) and common logarithms (log or {{fake link|log<sub>10</sub>|}}), but not all calculators have buttons for the logarithm of an arbitrary base. Accordingly, it is sometimes useful to change the base of a logarithm.{{pb}}As briefly mentioned in the {{fake link|§ Trivial identities}} section,  the definition of a logarithm is: <math>\log_b(y) = x \iff b^x = y.</math>  Solving for {{mvar|x}} in the exponential equation {{nobr|1={{mvar|b<sup>x</sup>}} = {{mvar|y}}}} could be done on most calculators by using common or natural logarithms: <math>b^x = y \iff x = \log_b(y) = \frac{\log(y)}{\log(b)} = \frac{\ln(y)}{\ln(b)}.</math>  For example, the [[binary logarithm]] ({{fake link|log<sub>2</sub>|}}), which is widely used in {{fake link|computer science}}, could be calculated on most calculators as: <math>2^x = y \iff x = \log_2(y) = \frac{\log(y)}{\log(2)} = \frac{\ln(y)}{\ln(2)}</math>{{pb}}More generally, the change of base formula can be formally defined as:<math>\forall a, b \in \mathbb{R}_+, a, b \neq 1,  \forall x \in \mathbb{R}_+, \log_b(x) = \frac{\log_a(x)}{\log_a(b)}</math>

=== Proof and derivation ===

Let <math>a, b \in \mathbb{R}_+</math>, where <math>a, b \neq 1</math> Let <math>x \in \mathbb{R}_+</math>. Here, <math>a</math> and <math>b</math> are the two bases we will be using for the logarithms. They cannot be 1, because the logarithm function is not well defined for the base of 1.{{Citation needed|date=July 2022}} The number <math>x</math> will be what the logarithm is evaluating, so it must be a positive number. Since we will be dealing with the term <math>\log_b(x)</math> quite frequently, we define it as a new variable: Let <math>m = \log_b(x)</math>.

To more easily manipulate the expression, it can be rewritten as an exponential.
<math display="block">b^m = x </math>

Applying <math>\log_a</math> to both sides of the equality,

<math display="block">\log_a(b^m) = \log_a(x) </math>

Now, using the logarithm of a power property, which states that <math>\log_a(b^m) = m\log_a(b)</math>,

<math display="block">m\log_a(b) = \log_a(x)</math>

Isolating <math>m</math>, we get the following:

<math display="block">m = \frac{\log_a(x)}{\log_a(b)}</math>

Resubstituting <math>m = \log_b(x)</math> back into the equation,

<math display="block">\log_b(x) = \frac{\log_a(x)}{\log_a(b)}</math>

This completes the proof that <math>\log_b(x) = \frac{\log_a(x)}{\log_a(b)}</math>.

This formula has several consequences:

<math display="block"> \log_b a = \frac 1 {\log_a b} </math>
<math display="block"> \log_{b^n} a =  {\log_b a \over n} </math>
<math display="block"> \log_{b} a =  \log_b e \cdot \log_e a = \log_b e \cdot \ln a </math>
<math display="block"> b^{\log_a d} = d^{\log_a b} </math>
<math display="block"> -\log_b a = \log_b \left({1 \over a}\right) = \log_{1/b} a</math>

<!-- extra blank space between two lines of "displayed" [[TeX]] for legibility -->

<math display="block"> \log_{b_1}a_1 \,\cdots\, \log_{b_n}a_n
= \log_{b_{\pi(1)}}a_1\, \cdots\, \log_{b_{\pi(n)}}a_n, </math>

where <math display="inline">\pi</math> is any [[permutation]] of the subscripts {{math|1, ..., ''n''}}.  For example

<math display="block"> \log_b w\cdot \log_a x \cdot \log_d c \cdot \log_d z 
= \log_d w \cdot \log_b x \cdot \log_a c \cdot \log_d z. </math>

=== Summation and subtraction ===
The following summation and subtraction rule is especially useful in [[probability theory]] when one is dealing with a sum of log-probabilities:

{|
|<math>\log_b (a+c) = \log_b a + \log_b \left(1 + \frac{c}{a}\right)</math>
|because
|<math>\left(a + c \right) = a \times \left(1 + \frac{c}{a} \right)</math>
|-
|<math>\log_b (a-c) = \log_b a + \log_b \left(1 - \frac{c}{a}\right)</math>
|because
|<math>\left(a - c \right) = a \times \left(1 - \frac{c}{a} \right)</math>
|}

Note that the subtraction identity is not defined if <math>a=c</math>, since the logarithm of zero is not defined. Also note that, when programming, <math>a</math> and <math>c</math> may have to be switched on the right hand side of the equations if <math>c \gg a</math> to avoid losing the "1 +" due to rounding errors.  Many programming languages have a specific <code>log1p(x)</code> function that calculates <math>\log_e (1+x)</math> without underflow (when <math>x</math> is small).

More generally:

<math display="block">\log _b \sum_{i=0}^N a_i = \log_b a_0 + \log_b \left( 1+\sum_{i=1}^N \frac{a_i}{a_0} \right) = \log _b a_0 + \log_b \left( 1+\sum_{i=1}^N b^{\left( \log_b a_i - \log _b a_0 \right)} \right)</math>

=== Exponents ===
A useful identity involving exponents:

<math display="block"> x^{\frac{\log(\log(x))}{\log(x)}} = \log(x)</math>

or more universally:

<math display="block"> x^{\frac{\log(a)}{\log(x)}} = a</math>

=== Other or resulting identities ===
<math display="block"> \frac{1}{\frac{1}{\log_x(a)} + \frac{1}{\log_y(a)}} = \log_{xy}(a)</math>
<math display="block"> \frac{1}{\frac{1}{\log_x(a)}-\frac{1}{\log_y(a)}} = \log_{\frac{x}{y}}(a)</math>

== Inequalities ==
Based on,<ref>{{citation
 | last = Topsøe | first = Flemming
 | editor1-last = Cho | editor1-first = Yeol Je
 | editor2-last = Kim | editor2-first = Jong Kyu
 | editor3-last = Dragomir | editor3-first = Sever S.
 | contribution = Some bounds for the logarithmic function
 | contribution-url = https://rgmia.org/papers/v7n2/pade.pdf
 | isbn = 978-1-59454-874-1
 | location = New York
 | mr = 2349596
 | pages = 137–151
 | publisher = Nova Science Publishers
 | title = Inequality theory and applications, Vol. 4: Papers from the 8th International Conference on Nonlinear Functional Analysis and Applications held at Gyeongsang National University, Chinju and Kyungnam University, Masan, August 9–13, 2004
 | year = 2007}}</ref>

<math display="block">\frac{x}{1+x} \leq \ln(1+x)
\leq \frac{x(6+x)}{6+4x}
\leq x \mbox{ for all } {-1} < x</math>
<math display="block">\begin{align}
\frac{2x}{2+x}&\leq3-\sqrt{\frac{27}{3+2x}}\leq\frac{x}{\sqrt{1+x+x^2/12}} \\[4pt]
&\leq    \ln(1+x)\leq \frac{x}{\sqrt{1+x}}\leq \frac{x}{2}\frac{2+x}{1+x} \\[4pt]
&\text{ for } 0 \le x \text{, reverse for } {-1} < x \le 0
\end{align}</math>

All are accurate around <math>x=0</math>, but not for large numbers.

== Tropical identities ==
The following identity relates [[log semiring]] to the [[min-plus semiring]].

<math display="block">\lim_{T \rightarrow 0} -T\log(e^{-\frac{s}{T}} + e^{-\frac{t}{T}}) = \mathrm{min}\{s,t\}</math>

== Calculus identities ==

=== Limits ===
<math display="block">\lim_{x\to 0^+}\log_a(x)=-\infty\quad \mbox{if } a > 1</math>
<math display="block">\lim_{x\to 0^+}\log_a(x)=\infty\quad \mbox{if } 0 < a < 1</math>
<math display="block">\lim_{x\to\infty}\log_a(x)=\infty\quad \mbox{if } a > 1</math>
<math display="block">\lim_{x\to\infty}\log_a(x)=-\infty\quad \mbox{if } 0 < a < 1</math>
<math display="block">\lim_{x\to \infty}x^b\log_a(x)=\infty\quad \mbox{if } b > 0</math>
<math display="block">\lim_{x\to\infty}\frac{\log_a(x)}{x^b}=0\quad \mbox{if } b > 0</math>

The last [[Limit of a function|limit]] is often summarized as "logarithms grow more slowly than any power or root of ''x''".

=== Derivatives of logarithmic functions ===
<math display="block">{d \over dx} \ln x = {1 \over x }, x > 0</math>
<math display="block">{d \over dx} \ln |x| = {1 \over x }, x \neq 0</math>
<math display="block">{d \over dx} \log_a x = {1 \over x \ln a}, x > 0, a > 0, \text{ and } a\neq 1</math>

=== Integral definition ===
<math display="block">\ln x = \int_1^x \frac {1}{t}\ dt </math>

To modify the limits of integration to run from <math>x</math> to <math>1</math>, we change the order of integration, which changes the sign of the integral:

<math display="block">-\int_1^x \frac{1}{t} \, dt = \int_x^1 \frac{1}{t} \, dt</math>

Therefore:

<math display="block">\ln \frac{1}{x} = \int_x^1 \frac{1}{t} \, dt</math>

=== Riemann Sum ===

<math display="block">\ln(n + 1) = </math>
<math display="block">\lim_{k \to \infty} \sum_{i=1}^{k} \frac{1}{x_i} \Delta x = </math>
<math display="block">\lim_{k \to \infty} \sum_{i=1}^{k} \frac{1}{1 + \frac{i-1}{k}n} \cdot \frac{n}{k} =</math>
<math display="block">\lim_{k \to \infty} \sum_{x=1}^{k \cdot n} \frac{1}{1 + \frac{x}{k}} \cdot \frac{1}{k} =</math>
<math display="block">\lim_{k \to \infty} \sum_{x=1}^{k \cdot n} \frac{1}{k + x} = \lim_{k \to \infty} \sum_{x=k+1}^{k \cdot n + k} \frac{1}{x} = \lim_{k \to \infty} \sum_{x=k+1}^{k (n + 1)} \frac{1}{x}</math>

for <math>\textstyle \Delta x = \frac{n}{k}</math> and <math>x_{i}</math> is a sample point in each interval.

=== Series representation ===

The natural logarithm <math>\ln(1 + x)</math> has a well-known [[Taylor series]]<ref>{{cite web
 | last = Weisstein
 | first = Eric W.
 | title = Mercator Series
 | website = MathWorld--A Wolfram Web Resource
 | url = https://mathworld.wolfram.com/MercatorSeries.html
 | access-date = 2024-04-24
}}</ref> expansion that converges for <math>x</math> in the [[Interval (mathematics)|open-closed interval]] {{open-closed|−1, 1}}:

<math display="block">\ln(1 + x) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1} x^n}{n} = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} - \frac{x^6}{6} + \cdots.</math>

Within this interval, for <math>x = 1</math>, the series is [[Conditional convergence|conditionally convergent]], and for all other values, it is [[Absolute convergence|absolutely convergent]]. For <math>x > 1</math> or <math>x \leq -1</math>, the series does not converge to <math>\ln(1 + x)</math>. In these cases, different representations<ref>To extend the utility of the Mercator series beyond its conventional bounds one can calculate <math display="inline"> \ln(1 + x)</math> for <math display="inline"> x = -\frac{n}{n+1}</math> and <math display="inline"> n \geq 0</math> and then [[#Using simpler operations|negate the result]], <math display="inline"> \ln\left(\frac{1}{n+1}\right)</math>, to derive <math display="inline"> \ln(n + 1)</math>. [[Natural logarithm of 2#Binary rising constant factorial|For example]], setting <math display="inline"> x = -\frac{1}{2}</math> yields <math display="inline"> \ln 2 = \sum_{n=1}^{\infty} \frac{1}{n 2^n}</math>.</ref> or methods must be used to evaluate the logarithm.

=== Harmonic number difference ===

It is not uncommon in advanced mathematics, particularly in [[analytic number theory]] and [[asymptotic analysis]], to encounter expressions involving differences or ratios of [[harmonic number]]s at scaled indices.<ref name="Flajolet2009AnalyticCombinatorics">{{cite book
 | last1 = Flajolet
 | first1 = Philippe
 | last2 = Sedgewick
 | first2 = Robert
 | title = Analytic Combinatorics
 | year = 2009
 | publisher = Cambridge University Press
 | isbn = 978-0521898065
 | page = 389
}} See page 117, and VI.8 definition of shifted harmonic numbers on page 389
</ref> The identity involving the limiting difference between harmonic numbers at scaled indices and its relationship to the logarithmic function provides an intriguing example of how discrete sequences can asymptotically relate to [[continuous functions]]. This identity is expressed as<ref name="Deveci2022DoubleSeries">{{cite arXiv
|last=Deveci
|first=Sinan
|title=On a Double Series Representation of the Natural Logarithm, the Asymptotic Behavior of Hölder Means, and an Elementary Estimate for the Prime Counting Function
|year=2022
|eprint=2211.10751
|class=math.NT
}} See Theorem 5.2. on pages 22 - 23</ref>

<math display="block">\lim_{{k \to \infty}} (H_{{k(n+1)}} - H_k) = \ln(n+1)</math>
which characterizes the behavior of harmonic numbers as they grow large. This approximation (which precisely equals <math>\ln(n+1)</math> in the limit) reflects how summation over increasing segments of the harmonic series exhibits [[Harmonic number#Calculation|integral properties]], giving insight into the interplay between discrete and continuous analysis. It also illustrates how understanding the behavior of sums and series at large scales can lead to insightful conclusions about their properties. Here <math>H_k</math> denotes the <math>k</math>-th harmonic number, defined as

<math display="block">H_k = \sum_{{j=1}}^k \frac{1}{j}</math>

The harmonic numbers are a fundamental sequence in number theory and analysis, known for their [[logarithmic growth]]. This result leverages the fact that the sum of the inverses of integers (i.e., harmonic numbers) can be closely approximated by the natural logarithm function, plus a [[Euler's constant|constant]], especially when extended over large intervals.<ref>{{cite book
 | last1 = Graham
 | first1 = Ronald L.
 | last2 = Knuth
 | first2 = Donald E.
 | last3 = Patashnik
 | first3 = Oren
 | title = Concrete Mathematics: A Foundation for Computer Science
 | year = 1994
 | publisher = Addison-Wesley
 | isbn = 0-201-55802-5
 | page = 429
}}
</ref><ref name="Flajolet2009AnalyticCombinatorics"/><ref>{{cite web
 | url = https://mathworld.wolfram.com/HarmonicNumber.html
 | title = Harmonic Number
 | publisher = Wolfram MathWorld
 | access-date = 2024-04-24
}} See formula 13.
</ref> As <math>k</math> tends towards infinity, the difference between the harmonic numbers <math>H_{k(n+1)}</math> and <math>H_k</math> converges to a non-zero value. This persistent non-zero difference, <math>\ln(n+1)</math>, precludes the possibility of the harmonic series approaching a finite limit, thus providing a clear mathematical articulation of its divergence.<ref>{{cite report
 | last1 = Kifowit
 | first1 = Steven J.
 | title = More Proofs of Divergence of the Harmonic Series
 | publisher = Prairie State College
 | year = 2019
 | url = https://stevekifowit.com/pubs/harm2.pdf
 | access-date = 2024-04-24
}} See Proofs 23 and 24 for details on the relationship between harmonic numbers and logarithmic functions.
</ref><ref>{{cite journal
 | last1 = Bell
 | first1 = Jordan
 | last2 = Blåsjö
 | first2 = Viktor
 | title = Pietro Mengoli's 1650 Proof That the Harmonic Series Diverges
 | journal = Mathematics Magazine
 | volume = 91
 | issue = 5
 | year = 2018
 | pages = 341–347
 | doi = 10.1080/0025570X.2018.1506656
 | jstor = 48665556
 | hdl = 1874/407528
 | url = https://www.jstor.org/stable/48665556
 | access-date = 2024-04-24
| hdl-access = free
 }}
</ref> The technique of approximating sums by integrals (specifically using the [[Integral test for convergence|integral test]] or by [[direct integral]] approximation) is fundamental in deriving such results. This specific identity can be a consequence of these approximations, considering:

<math display="block">\sum_{{j=k+1}}^{k(n+1)} \frac{1}{j} \approx \int_k^{k(n+1)} \frac{dx}{x}</math>

==== Harmonic limit derivation ====

The limit explores the growth of the harmonic numbers when indices are multiplied by a scaling factor and then differenced. It specifically captures the sum from <math>k+1</math> to <math>k(n+1)</math>:

<math display="block">H_{{k(n+1)}} - H_k = \sum_{{j=k+1}}^{k(n+1)} \frac{1}{j}</math>

This can be estimated using the integral test for convergence, or more directly by comparing it to the [[#Riemann Sum|integral]] of <math>1/x</math> from <math>k</math> to <math>k(n+1)</math>:

<math display="block">\lim_{{k \to \infty}} \sum_{{j=k+1}}^{k(n+1)} \frac{1}{j} = \int_k^{k(n+1)} \frac{dx}{x} = \ln(k(n+1)) - \ln(k) = \ln\left(\frac{k(n+1)}{k}\right) = \ln(n+1)</math>

As the window's lower bound begins at <math>k+1</math> and the upper bound extends to <math>k(n+1)</math>, both of which tend toward infinity as <math>k \to \infty</math>, the summation window encompasses an increasingly vast portion of the smallest possible terms of the harmonic series (those with astronomically large denominators), creating a discrete sum that stretches towards infinity, which mirrors how continuous integrals accumulate value across an infinitesimally fine partitioning of the domain. In the limit, the interval is effectively from <math>1</math> to <math>n+1</math> where the onset <math>k</math> implies this minimally discrete region.

=== Double series formula ===
The [[#Harmonic number difference|harmonic number difference]] formula for <math>\ln(m)</math> is an extension<ref name="Deveci2022DoubleSeries"/> of the classic, [[Harmonic series (mathematics)#Alternating harmonic series|alternating identity]] of <math>\ln(2)</math>:

<math display="block">\ln(2) = \lim_{k \to \infty} \sum_{n=1}^{k} \left( \frac{1}{2n-1} - \frac{1}{2n} \right)</math>

which can be generalized as the double series over the [[Modular_arithmetic#Integers_modulo_m|residues]] of <math>m</math>:

<math display="block">\ln(m) = \sum_{x \in \langle m \rangle \cap \mathbb{N}} \sum_{r \in \mathbb{Z}_m \cap \mathbb{N}} \left( \frac{1}{x-r} - \frac{1}{x} \right) = \sum_{x \in \langle m \rangle \cap \mathbb{N}} \sum_{r \in \mathbb{Z}_m \cap \mathbb{N}} \frac{r}{x(x-r)}
</math>

where <math>\langle m \rangle</math> is the [[Principal_ideal|principal ideal]] generated by <math>m</math>. Subtracting <math>\textstyle \frac{1}{x}</math> from each term <math>\textstyle \frac{1}{x-r}</math> (i.e., balancing each term with the modulus) reduces the magnitude of each term's contribution, ensuring [[Alternating series test|convergence]] by controlling the series' tendency toward divergence as <math>m</math> increases. For example:

<math display="block">\ln(4) = \lim_{k \to \infty} \sum_{n=1}^{k} \left( \frac{1}{4n-3} - \frac{1}{4n} \right) + \left( \frac{1}{4n-2} - \frac{1}{4n} \right) + \left( \frac{1}{4n-1} - \frac{1}{4n} \right)</math>

This method leverages the fine differences between closely related terms to stabilize the series. The sum over all residues <math>r \in \N</math> ensures that adjustments are uniformly applied across all possible offsets within each block of <math>m</math> terms. This uniform distribution of the "correction" across different intervals defined by <math>x-r</math> functions similarly to [[Telescoping series|telescoping]] over a very large sequence. It helps to flatten out the discrepancies that might otherwise lead to divergent behavior in a straightforward harmonic series. Note that the structure of the [[Addition#summand|summands]] of this formula matches those of the [[Harmonic number#Alternative, asymptotic formulation|interpolated harmonic number]] <math>H_x</math> when both the [[Domain of a function|domain]] and [[Range of a function|range]] are negated (i.e., <math>-H_{-x}</math>). However, the interpretation and roles of the variables differ.

==== Deveci's Proof ====
A fundamental feature of the proof is the accumulation of the [[Subtraction#Notation and terminology|subtrahends]] <math display="inline"> \frac{1}{x}</math> into a [[unit fraction]], that is, <math display="inline"> \frac{m}{x} = \frac{1}{n}</math> for <math>m \mid x</math>, thus <math>m = \omega + 1</math> rather than <math>m = |\mathbb{Z}_{m} \cap \mathbb{N}|</math>, where the [[Maximum and minimum|extrema]] of <math>\mathbb{Z}_{m} \cap \mathbb{N}</math> are <math>[0, \omega]</math> if <math>\mathbb{N} = \mathbb{N}_{0}</math> and <math>[1, \omega]</math> [[Natural number#Notation|otherwise]], with the minimum of <math>0</math> being implicit in the latter case due to the structural requirements of the proof. Since the [[Cardinality#Comparing sets|cardinality]] of <math>\mathbb{Z}_{m} \cap \mathbb{N}</math> depends on the selection of one of two possible minima, the integral <math>\textstyle \int \frac{1}{t} dt</math>, as a set-theoretic procedure, is a function of the maximum <math>\omega</math> (which remains consistent across both interpretations) plus <math>1</math>, not the cardinality (which is ambiguous<ref>{{Cite arXiv |eprint=1102.0418 |first=Peter |last=Harremoës |title=Is Zero a Natural Number? |year=2011|class=math.HO }} A synopsis on the nature of 0 which frames the choice of minimum as the dichotomy between ordinals and cardinals.</ref><ref>{{Cite journal |last=Barton |first=N. |year=2020 |title=Absence perception and the philosophy of zero |journal=Synthese |volume=197 |issue=9 |pages=3823–3850 |doi=10.1007/s11229-019-02220-x |pmc=7437648 |pmid=32848285}} See section 3.1</ref> due to varying definitions of the minimum). Whereas the harmonic number difference computes the integral in a global sliding window, the double series, in parallel, computes the sum in a local sliding window—a shifting <math>m</math>-tuple—over the harmonic series, advancing the window by <math>m</math> positions to select the next <math>m</math>-tuple, and offsetting each element of each tuple by <math display="inline"> \frac{1}{m}</math> relative to the window's absolute position. The sum <math display="inline">\sum_{n=1}^{k} \sum \frac{1}{x - r}</math> corresponds to <math>H_{km}</math> which scales <math>H_{m}</math> without bound. The sum <math display="inline"> \sum_{n=1}^{k} -\frac{1}{n}</math> corresponds to the prefix <math>H_{k}</math> trimmed from the series to establish the window's moving lower bound <math>k+1</math>, and <math>\ln(m)</math> is the limit of the sliding window (the scaled, truncated<ref>The <math>k+1</math> shift is characteristic of the [[#Riemann Sum|right Riemann sum]] employed to prevent the integral from degenerating into the harmonic series, thereby averting divergence. Here, <math display="inline"> -\frac{1}{n}</math> functions analogously, serving to regulate the series. The successor operation <math>m = \omega + 1</math> signals the implicit inclusion of the modulus <math>m</math> (the region omitted from <math>\mathbb{N}_{1}</math>). The importance of this, from an axiomatic perspective, becomes evident when the residues of <math>m</math> are formulated as <math>e^{\ln(\omega + 1)}</math>, where <math>\omega + 1</math> is bootstrapped by <math>\omega = 0</math> to produce the residues of modulus <math>m = \omega = \omega_{0} + 1 = 1</math>. Consequently, <math>\omega</math> represents a limiting value in this context.</ref> series):

<math display="block">\begin{align}
\sum_{n=1}^k \sum_{r=1}^{\omega} \left( \frac{1}{mn - r} - \frac{1}{mn} \right)
&= \sum_{n=1}^k \sum_{r=0}^{\omega} \left( \frac{1}{mn - r} - \frac{1}{mn} \right) \\
&= \sum_{n=1}^k \left( -\frac{1}{n} + \sum_{r=0}^{\omega} \frac{1}{mn - r} \right) \\
&= -H_k + \sum_{n=1}^k \sum_{r=0}^{\omega} \frac{1}{mn - r} \\
&= -H_k + \sum_{n=1}^k \sum_{r=0}^{\omega} \frac{1}{(n-1)m + m - r} \\
&= -H_k + \sum_{n=1}^k \sum_{j=1}^m \frac{1}{(n-1)m + j} \\
&= -H_k + \sum_{n=1}^k \left( H_{nm} - H_{m(n-1)} \right) \\
&= -H_k + H_{mk}
\end{align}</math>
<math display="block">\lim_{k \to \infty} H_{km} - H_k = \sum_{x \in \langle m \rangle \cap \mathbb{N}} \sum_{r \in \mathbb{Z}_m \cap \mathbb{N}} \left( \frac{1}{x-r} - \frac{1}{x} \right) = \ln(\omega + 1) = \ln(m)</math>

=== Integrals of logarithmic functions ===
<math display="block">\int \ln x \, dx = x \ln x - x + C = x(\ln x - 1) + C</math>
<math display="block">\int \log_a x \, dx = x \log_a x - \frac{x}{\ln a} + C = \frac{x (\ln x - 1)}{\ln a} + C</math>

To remember higher [[integral]]s, it is convenient to define:

<math display="block">x^{\left [n \right]} = x^{n}(\log(x) - H_n)</math>

where <math>H_n</math> is the ''n''<sup>th</sup> [[harmonic number]]:

<math display="block">x^{\left [ 0 \right ]} = \log x</math>
<math display="block">x^{\left [ 1 \right ]} = x \log(x) - x</math>
<math display="block">x^{\left [ 2 \right ]} = x^2 \log(x) - \begin{matrix} \frac{3}{2} \end{matrix}x^2</math>
<math display="block">x^{\left [ 3 \right ]} = x^3 \log(x) - \begin{matrix} \frac{11}{6} \end{matrix}x^3</math>

Then:

<math display="block">\frac{d}{dx}\, x^{\left[ n \right]} = nx^{\left[ n-1 \right]}</math>
<math display="block">\int x^{\left[ n \right]}\,dx = \frac{x^{\left [ n+1 \right ]}}{n+1} + C</math>

== Approximating large numbers ==

The identities of logarithms can be used to approximate large numbers. Note that {{math|1=log<sub>''b''</sub>(''a'') + log<sub>''b''</sub>(''c'') = log<sub>''b''</sub>(''ac'')}}, where ''a'', ''b'', and ''c'' are arbitrary constants. Suppose that one wants to approximate the 44th [[Mersenne prime]], {{math|2<sup>32,582,657</sup> &minus;1}}. To get the base-10 logarithm, we would multiply 32,582,657 by {{math|log<sub>10</sub>(2)}}, getting {{math|1=9,808,357.09543 = 9,808,357 + 0.09543}}. We can then get {{math|1=10<sup>9,808,357</sup> &times; 10<sup>0.09543</sup> ≈ 1.25 &times; 10<sup>9,808,357</sup>}}.

Similarly, [[factorial]]s can be approximated by summing the logarithms of the terms.

== Complex logarithm identities ==

The [[complex logarithm]] is the [[complex number]] analogue of the logarithm function. No single valued function on the complex plane can satisfy the normal rules for logarithms. However, a [[multivalued function]] can be defined which satisfies most of the identities. It is usual to consider this as a function defined on a [[Riemann surface]]. A single valued version, called the [[principal value]] of the logarithm, can be defined which is discontinuous on the negative x axis, and is equal to the multivalued version on a single [[branch cut]].

=== Definitions ===

In what follows, a capital first letter is used for the principal value of functions, and the lower case version is used for the multivalued function. The single valued version of definitions and identities is always given first, followed by a separate section for the multiple valued versions.

*{{math|ln(''r'')}} is the standard [[natural logarithm]] of the real number {{mvar|r}}.
*{{math|Arg(''z'')}} is the principal value of the [[Arg (mathematics)|arg]] function; its value is restricted to {{open-closed|−''π'', ''π''}}. It can be computed using {{math|1=Arg(''x'' + ''iy'') = [[atan2]](''y'', ''x'')}}.
*{{math|Log(''z'')}} is the principal value of the complex logarithm function and has imaginary part in the range {{open-closed|−''π'', ''π''}}.
*<math>\operatorname{Log}(z) = \ln(|z|) + i \operatorname{Arg}(z)</math>
*<math>e^{\operatorname{Log}(z)} = z</math>

The multiple valued version of {{math|log(''z'')}} is a set, but it is easier to write it without braces and using it in formulas follows obvious rules.

*{{math|log(''z'')}} is the set of complex numbers ''v'' which satisfy {{math|1=e<sup>''v''</sup> = ''z''}}
*{{math|arg(''z'')}} is the set of possible values of the [[Arg (mathematics)|arg]] function applied to ''z''.

When ''k'' is any integer:

<math display="block">\log(z) = \ln(|z|) + i \arg(z)</math>
<math display="block">\log(z) = \operatorname{Log}(z) + 2 \pi i k</math>
<math display="block">e^{\log(z)} = z</math>

=== Constants ===

Principal value forms:

<math display="block">\operatorname{Log}(1) = 0</math>
<math display="block">\operatorname{Log}(e) = 1</math>

Multiple value forms, for any ''k'' an integer:

<math display="block">\log(1) = 0 + 2 \pi i k</math>
<math display="block">\log(e) = 1 + 2 \pi i k</math>

=== Summation ===

Principal value forms:<ref name=":0">{{Cite book|last=Abramowitz|first=Milton|title=Handbook of mathematical functions, with formulas, graphs, and mathematical tables|date=1965|publisher=Dover Publications|others=Irene A. Stegun|isbn=0-486-61272-4|location=New York|oclc=429082}}</ref>

<math display="block">\operatorname{Log}(z_1) + \operatorname{Log}(z_2) = \operatorname{Log}(z_1 z_2) \pmod {2 \pi i}</math>
<math display="block">\operatorname{Log}(z_1) + \operatorname{Log}(z_2) = \operatorname{Log}(z_1 z_2)\quad 
(-\pi <\operatorname{Arg}(z_1)+\operatorname{Arg}(z_2)\leq \pi; 
\text{ e.g., } \operatorname{Re}z_1\geq 0 \text{ and } \operatorname{Re}z_2 > 0)</math>
<math display="block">\operatorname{Log}(z_1) - \operatorname{Log}(z_2) = \operatorname{Log}(z_1 / z_2) \pmod {2 \pi i}</math>
<math display="block">\operatorname{Log}(z_1) - \operatorname{Log}(z_2) = \operatorname{Log}(z_1 / z_2) 
\quad 
(-\pi <\operatorname{Arg}(z_1)-\operatorname{Arg}(z_2)\leq \pi; 
\text{ e.g., } \operatorname{Re}z_1\geq 0 \text{ and } \operatorname{Re}z_2 > 0)</math>

Multiple value forms:

<math display="block">\log(z_1) + \log(z_2) = \log(z_1 z_2)</math>
<math display="block">\log(z_1) - \log(z_2) = \log(z_1 / z_2)</math>

=== Powers ===

A complex power of a complex number can have many possible values.

Principal value form:

<math display="block">{z_1}^{z_2} = e^{z_2 \operatorname{Log}(z_1)} </math>
<math display="block">\operatorname{Log}{\left({z_1}^{z_2}\right)} = z_2 \operatorname{Log}(z_1) \pmod {2 \pi i}</math>

Multiple value forms:

<math display="block">{z_1}^{z_2} = e^{z_2 \log(z_1)}</math>

Where {{math|''k''<sub>1</sub>}}, {{math|''k''<sub>2</sub>}} are any integers:

<math display="block">\log{\left({z_1}^{z_2}\right)} = z_2 \log(z_1) + 2 \pi i k_2</math>
<math display="block">\log{\left({z_1}^{z_2}\right)} = z_2 \operatorname{Log}(z_1) + z_2 2 \pi i k_1 + 2 \pi i k_2</math>

== Asymptotic identities ==
=== Pronic numbers ===
As a consequence of the [[#Harmonic number difference|harmonic number difference]], the natural logarithm is [[Asymptotic analysis|asymptotically]] approximated by a finite [[Series (mathematics)|series]] difference,<ref name="Deveci2022DoubleSeries"/> representing a truncation of the [[#Riemann Sum|integral]] at <math>k = n</math>:

<math display="block">H_{{2T[n]}} - H_n \sim \ln(n+1)</math>

where <math>T[n]</math> is the {{mvar|n}}th [[triangular number]], and <math>2T[n]</math> is the sum of the [[Pronic number#As figurate numbers|first {{mvar|n}} even integers]]. Since the {{mvar|n}}th [[pronic number]] is asymptotically equivalent to the {{mvar|n}}th [[Square number|perfect square]], it follows that:

<math display="block">H_{{n^2}} - H_n \sim \ln(n+1)</math>

=== Prime number theorem===
The [[prime number theorem]] provides the following asymptotic equivalence:

<math display="block">\frac{n}{\pi(n)} \sim \ln n</math>

where <math>\pi(n)</math> is the [[prime counting function]]. This relationship is equal to:<ref name="Deveci2022DoubleSeries"/>{{rp|2}}

<math display="block">\frac{n}{H(1, 2, \ldots, x_n)} \sim \ln n</math>

where <math>H(x_1, x_2, \ldots, x_n)</math> is the [[harmonic mean]] of <math>x_1, x_2, \ldots, x_n</math>. This is derived from the fact that the difference between the <math>n</math>th harmonic number and <math>\ln n</math> asymptotically approaches a [[Euler's constant|small constant]], resulting in <math>H_{n^2} - H_n \sim H_n</math>. This behavior can also be derived from the [[#Logarithm of a power|properties of logarithms]]: <math>\ln n</math> is half of <math>\ln n^2</math>, and this "first half" is the natural log of the root of <math>n^2</math>, which corresponds roughly to the first <math>\textstyle \frac{1}{n}</math>th of the sum <math>H_{n^2}</math>, or <math>H_n</math>. The asymptotic equivalence of the first <math>\textstyle \frac{1}{n}</math>th of <math>H_{n^2}</math> to the latter <math>\textstyle \frac{n-1}{n}</math>th of the series is expressed as follows:

<math display="block">\frac{H_n}{H_{n^2}} \sim \frac{\ln \sqrt{n}}{\ln n} = \frac{1}{2}</math>

which generalizes to:

<math display="block">\frac{H_n}{H_{n^k}} \sim \frac{\ln \sqrt[k]{n}}{\ln n} = \frac{1}{k}</math>
<math display="block">k H_n \sim H_{n^k}</math>

and:

<math display="block">k H_n - H_n \sim (k - 1) \ln(n+1)</math>
<math display="block">H_{{n^k}} - H_n \sim (k - 1) \ln(n+1)</math>
<math display="block">k H_n - H_{{n^k}} \sim (k - 1) \gamma</math>

for fixed <math>k</math>. The correspondence sets <math>H_n</math> as a [[Unit of measurement|unit magnitude]] that partitions <math>H_{n^k}</math> across powers, where each interval <math>\textstyle \frac{1}{n}</math> to <math>\textstyle \frac{1}{n^2}</math>, <math>\textstyle \frac{1}{n^2}</math> to <math>\textstyle \frac{1}{n^3}</math>, etc., corresponds to one <math>H_n</math> unit, illustrating that <math>H_{n^k}</math> forms a [[Harmonic series (mathematics)#Definition and divergence|divergent series]] as <math>k \to \infty</math>.

=== Real Arguments ===
These approximations extend to the real-valued domain through the [[Harmonic number#Alternative, asymptotic formulation|interpolated harmonic number]]. For example, where <math>x \in \mathbb{R}</math>:

<math display="block">H_{{x^2}} - H_x \sim \ln x</math>

=== Stirling numbers ===
The natural logarithm is asymptotically related to the harmonic numbers by the [[Stirling number]]s<ref>{{cite arXiv 
| title = New series identities with Cauchy, Stirling, and harmonic numbers, and Laguerre polynomials
| author = Khristo N. Boyadzhiev
| year = 2022
| eprint = 1911.00186
| pages = 2, 6
| class = math.NT
}}</ref> and the [[Gregory coefficients]].<ref>{{cite book 
| last = Comtet 
| first = Louis 
| title = Advanced Combinatorics 
| publisher = Kluwer 
| year = 1974
}}</ref> By representing <math>H_n</math> in terms of [[Stirling numbers of the first kind]], the harmonic number difference is alternatively expressed as follows, for fixed <math>k</math>:

<math display="block">\frac{s(n^k+1, 2)}{(n^k)!} - \frac{s(n+1, 2)}{n!} \sim (k-1) \ln(n+1)</math>

== See also ==

* {{annotated link|List of formulae involving π}}
* {{annotated link|List of integrals of logarithmic functions}}
* {{annotated link|List of mathematical identities}}
* {{annotated link|Lists of mathematics topics}}
* {{annotated link|List of trigonometric identities}}

== References ==
{{reflist}}

== External links ==
*{{sister-inline
|project=v
|links=[[v:Speak Math Now!/Week 9: Six rules of Exponents/Logarithms|A lesson on logarithms can be found on Wikiversity]]
|short=yes
}}
* [https://www.mathwords.com/l/logarithm.htm Logarithm] in Mathwords

[[Category:Logarithms]] 
[[Category:Mathematical identities]]
[[Category:Articles containing proofs]]