Natural logarithm

The natural logarithm of a number is its logarithm to the base of the mathematical constant $e$ , which is an irrational and transcendental number approximately equal to $2.718 281828459$ .^[1] The natural logarithm of $x$ is generally written as $ln x$ , $log e x$ , or sometimes, if the base $e$ is implicit, simply $log x$ .^[2]^[3] Parentheses are sometimes added for clarity, giving $ln(x)$ , $log e (x)$ , or $log(x)$ . This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity.

"Base e" redirects here. For the numbering system which uses "e" as its base, see Non-integer base of numeration § Base e.

Natural logarithm

$\ln x=\log _{e}x$

Analytic proofs

Pure and applied mathematics

$\mathbb {R} _{>0}$

$\mathbb {R}$

+∞

1

$x=0$

1

$\exp x$

${\dfrac {d}{dx}}\ln x={\dfrac {1}{x}},x>0$

$\int \ln x\,dx=x\left(\ln x-1\right)+C$

The natural logarithm of $x$ is the power to which $e$ would have to be raised to equal $x$ . For example, $ln 7.5$ is $2.0149...$ , because $e 2.0149... = 7.5$ . The natural logarithm of $e$ itself, $ln e$ , is $1$ , because $e 1 = e$ , while the natural logarithm of $1$ is $0$ , since $e 0 = 1$ .

The natural logarithm can be defined for any positive real number $a$ as the area under the curve $y = 1/ x$ from $1$ to $a$ ^[4] (with the area being negative when $0 < a < 1$ ). The simplicity of this definition, which is matched in many other formulas involving the natural logarithm, leads to the term "natural". The definition of the natural logarithm can then be extended to give logarithm values for negative numbers and for all non-zero complex numbers, although this leads to a multi-valued function: see complex logarithm for more.

The natural logarithm function, if considered as a real-valued function of a positive real variable, is the inverse function of the exponential function, leading to the identities: ${\begin{aligned}e^{\ln x}&=x\qquad {\text{ if }}x\in \mathbb {R} _{+}\\\ln e^{x}&=x\qquad {\text{ if }}x\in \mathbb {R} \end{aligned}}$

Like all logarithms, the natural logarithm maps multiplication of positive numbers into addition:^[5] $\ln(x\cdot y)=\ln x+\ln y~.$

Logarithms can be defined for any positive base other than 1, not only $e$ . However, logarithms in other bases differ only by a constant multiplier from the natural logarithm, and can be defined in terms of the latter, $\log _{b}x=\ln x/\ln b=\ln x\cdot \log _{b}e$ .

Logarithms are useful for solving equations in which the unknown appears as the exponent of some other quantity. For example, logarithms are used to solve for the half-life, decay constant, or unknown time in exponential decay problems. They are important in many branches of mathematics and scientific disciplines, and are used to solve problems involving compound interest.

Notational conventions[edit]

The notations $ln x$ and $log e x$ both refer unambiguously to the natural logarithm of $x$ , and $log x$ without an explicit base may also refer to the natural logarithm. This usage is common in mathematics, along with some scientific contexts as well as in many programming languages.^{[nb 1]} In some other contexts such as chemistry, however, $log x$ can be used to denote the common (base 10) logarithm. It may also refer to the binary (base 2) logarithm in the context of computer science, particularly in the context of time complexity.

$\ln 1=0$

$\ln e=1$

$\ln(xy)=\ln x+\ln y\quad {\text{for }}\;x>0\;{\text{and }}\;y>0$

$\ln(x/y)=\ln x-\ln y\quad {\text{for }}\;x>0\;{\text{and }}\;y>0$

$\ln(x^{y})=y\ln x\quad {\text{for }}\;x>0$

$\ln({\sqrt[{y}]{x}})=(\ln x)/y\quad {\text{for }}\;x>0\;{\text{and }}\;y\neq 0$

$\ln x<\ln y\quad {\text{for }}\;0<x<y$

$\lim _{x\to 0}{\frac {\ln(1+x)}{x}}=1$

$\lim _{\alpha \to 0}{\frac {x^{\alpha }-1}{\alpha }}=\ln x\quad {\text{for }}\;x>0$

${\frac {x-1}{x}}\leq \ln x\leq x-1\quad {\text{for}}\quad x>0$

$\ln {(1+x^{\alpha })}\leq \alpha x\quad {\text{for}}\quad x\geq 0\;{\text{and }}\;\alpha \geq 1$

The natural logarithm has the following mathematical properties:

Derivative[edit]

The derivative of the natural logarithm as a real-valued function on the positive reals is given by^[4] ${\frac {d}{dx}}\ln x={\frac {1}{x}}.$

How to establish this derivative of the natural logarithm depends on how it is defined firsthand. If the natural logarithm is defined as the integral $\ln x=\int _{1}^{x}{\frac {1}{t}}\,dt,$ then the derivative immediately follows from the first part of the fundamental theorem of calculus.

On the other hand, if the natural logarithm is defined as the inverse of the (natural) exponential function, then the derivative (for $x > 0$ ) can be found by using the properties of the logarithm and a definition of the exponential function.

From the definition of the number $e=\lim _{u\to 0}(1+u)^{1/u},$ the exponential function can be defined as $e^{x}=\lim _{u\to 0}(1+u)^{x/u}=\lim _{h\to 0}(1+hx)^{1/h},$ where $u=hx,h={\frac {u}{x}}.$

The derivative can then be found from first principles. ${\begin{aligned}{\frac {d}{dx}}\ln x&=\lim _{h\to 0}{\frac {\ln(x+h)-\ln x}{h}}\\&=\lim _{h\to 0}\left[{\frac {1}{h}}\ln \left({\frac {x+h}{x}}\right)\right]\\&=\lim _{h\to 0}\left[\ln \left(1+{\frac {h}{x}}\right)^{\frac {1}{h}}\right]\quad &&{\text{all above for logarithmic properties}}\\&=\ln \left[\lim _{h\to 0}\left(1+{\frac {h}{x}}\right)^{\frac {1}{h}}\right]\quad &&{\text{for continuity of the logarithm}}\\&=\ln e^{1/x}\quad &&{\text{for the definition of }}e^{x}=\lim _{h\to 0}(1+hx)^{1/h}\\&={\frac {1}{x}}\quad &&{\text{for the definition of the ln as inverse function.}}\end{aligned}}$

Also, we have: ${\frac {d}{dx}}\ln ax={\frac {d}{dx}}(\ln a+\ln x)={\frac {d}{dx}}\ln a+{\frac {d}{dx}}\ln x={\frac {1}{x}}.$

so, unlike its inverse function $e^{ax}$ , a constant in the function doesn't alter the differential.

Continued fractions[edit]

While no simple continued fractions are available, several generalized continued fractions exist, including: ${\begin{aligned}\ln(1+x)&={\frac {x^{1}}{1}}-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-{\frac {x^{4}}{4}}+{\frac {x^{5}}{5}}-\cdots \\[5pt]&={\cfrac {x}{1-0x+{\cfrac {1^{2}x}{2-1x+{\cfrac {2^{2}x}{3-2x+{\cfrac {3^{2}x}{4-3x+{\cfrac {4^{2}x}{5-4x+\ddots }}}}}}}}}}\end{aligned}}$ ${\begin{aligned}\ln \left(1+{\frac {x}{y}}\right)&={\cfrac {x}{y+{\cfrac {1x}{2+{\cfrac {1x}{3y+{\cfrac {2x}{2+{\cfrac {2x}{5y+{\cfrac {3x}{2+\ddots }}}}}}}}}}}}\\[5pt]&={\cfrac {2x}{2y+x-{\cfrac {(1x)^{2}}{3(2y+x)-{\cfrac {(2x)^{2}}{5(2y+x)-{\cfrac {(3x)^{2}}{7(2y+x)-\ddots }}}}}}}}\end{aligned}}$

These continued fractions—particularly the last—converge rapidly for values close to 1. However, the natural logarithms of much larger numbers can easily be computed, by repeatedly adding those of smaller numbers, with similarly rapid convergence.

For example, since 2 = 1.25³ × 1.024, the natural logarithm of 2 can be computed as: ${\begin{aligned}\ln 2&=3\ln \left(1+{\frac {1}{4}}\right)+\ln \left(1+{\frac {3}{125}}\right)\\[8pt]&={\cfrac {6}{9-{\cfrac {1^{2}}{27-{\cfrac {2^{2}}{45-{\cfrac {3^{2}}{63-\ddots }}}}}}}}+{\cfrac {6}{253-{\cfrac {3^{2}}{759-{\cfrac {6^{2}}{1265-{\cfrac {9^{2}}{1771-\ddots }}}}}}}}.\end{aligned}}$

Furthermore, since 10 = 1.25¹⁰ × 1.024³, even the natural logarithm of 10 can be computed similarly as: ${\begin{aligned}\ln 10&=10\ln \left(1+{\frac {1}{4}}\right)+3\ln \left(1+{\frac {3}{125}}\right)\\[10pt]&={\cfrac {20}{9-{\cfrac {1^{2}}{27-{\cfrac {2^{2}}{45-{\cfrac {3^{2}}{63-\ddots }}}}}}}}+{\cfrac {18}{253-{\cfrac {3^{2}}{759-{\cfrac {6^{2}}{1265-{\cfrac {9^{2}}{1771-\ddots }}}}}}}}.\end{aligned}}$ The reciprocal of the natural logarithm can be also written in this way: ${\frac {1}{\ln(x)}}={\frac {2x}{x^{2}-1}}{\sqrt {{\frac {1}{2}}+{\frac {x^{2}+1}{4x}}}}{\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {{\frac {1}{2}}+{\frac {x^{2}+1}{4x}}}}}}\ldots$

For example: ${\frac {1}{\ln(2)}}={\frac {4}{3}}{\sqrt {{\frac {1}{2}}+{\frac {5}{8}}}}{\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {{\frac {1}{2}}+{\frac {5}{8}}}}}}\ldots$

Plots of the natural logarithm function on the complex plane ()

principal branch

$z = Re(ln(x + yi))$

$z = | (Im(ln(x + yi))) |$

$z = | (ln(x + yi)) |$

Superposition of the previous three graphs

The exponential function can be extended to a function which gives a complex number as $e z$ for any arbitrary complex number $z$ ; simply use the infinite series with $x$ =z complex. This exponential function can be inverted to form a complex logarithm that exhibits most of the properties of the ordinary logarithm. There are two difficulties involved: no $x$ has $e x = 0$ ; and it turns out that $e 2 iπ = 1 = e 0$ . Since the multiplicative property still works for the complex exponential function, $e z = e z +2 kiπ$ , for all complex $z$ and integers $k$ .

So the logarithm cannot be defined for the whole complex plane, and even then it is multi-valued—any complex logarithm can be changed into an "equivalent" logarithm by adding any integer multiple of $2 iπ$ at will. The complex logarithm can only be single-valued on the cut plane. For example, $ln i = iπ / 2$ or $5 iπ / 2$ or $- 3 iπ / 2$ , etc.; and although $i 4 = 1, 4 ln i$ can be defined as $2 iπ$ , or $10 iπ$ or $-6 iπ$ , and so on.

Natural logarithm