Logarithm and Logarithmic Function¶

Definition¶

The logarithm of a positive real number $x$ with respect to base $b$ ($b>$$b>0$ and $b\neq 1$$1$) is the exponent to which $b$ must be raised to get $x\text{.}$ In other words, the logarithm of $x$ to base $b$ is the solution $y$ to the equation $b^y=x\text{.}$ This logarithm is denoted by $\log_b x\text{.}$ The function $g(x)=\log_b x$ on $\mathbb{R}^+$ is called the logarithmic function of $x$ to base $b\text{.}$ It is the inverse of the exponential function with base $b\text{.}$ The logarithm with base $\mathrm{e}$ is called the natural logarithm, denoted $\ln x\text{.}$ Without a base, $\log x$ usually means $\ln x$ in analysis texts but $\log_{10}x$ in some other texts, so it is safest to state the base.

Interpretation¶

The logarithm answers the question "which power?": $\log_b x$ is the exponent $y$ with $b^y=x$$x$. Logarithmic functions therefore measure exponents and invert exponential growth, and they transform multiplicative relationships into additive ones.

$\ln$ undoes $\mathrm{e}^x$: the points $(0,1)$, $(1,\mathrm{e})$ on $y=\mathrm{e}^x$ mirror to $(1,0)$, $(\mathrm{e},1)$ on $y=\ln x$$\ln x$, so $\ln 1=0$ and $\ln \mathrm{e}=1$$1$. The reflection in the dashed line $y=x$ swaps inputs and outputs, so the logarithm accepts only positive inputs but can return any real number.

The restrictions in the definition come from the exponential function. For $b>0$ with $b\neq1$$1$, every value $b^y$ is positive and every positive number is reached exactly once, so $\log_b x$ exists only for $x>0$ and is then unique. The base $b=1$ is excluded because $1^y=1$ for every $y$: the equation $1^y=x$ has no solution for $x\neq1$ and infinitely many for $x=1$$1$. For $b>1$ the logarithm is negative on $(0,1)$, zero at $x=1$ and positive beyond.

The product law is the reason logarithms turn growth factors into increments: multiplying $x$ by $b$ adds exactly $1$ to $\log_b x$. Different bases differ only by a constant factor, $\log_b x=\ln x/\ln b$$\ln x/\ln b$, so the natural logarithm, with base Euler's number, is enough in practice.

A common slip is to apply the product law to sums: $\log_b(x+y)$ is in general not $\log_b x+\log_b y$${\log }_{b}y$. For instance, $\log_2(1+1)=1$$1$, whereas $\log_2 1+\log_2 1=0$$0$.

Key Results¶

• Logarithms are defined only for positive inputs.
• Logarithmic functions are strictly monotone.
• We have the following results:
  • Product law: $\displaystyle \log_b(xy)=\log_b x+\log_b y\text{;}$
  • Quotient law: $\displaystyle \log_b(x/y)=\log_b x-\log_b y\text{;}$
  • Power law: $\displaystyle \log_b\left(x^\alpha\right)=\alpha\log_b x\text{.}$

Examples¶

• $\ln\mathrm{e}=1\text{.}$
• $\log_2 8=3\text{.}$

Prerequisites¶

• Exponential Function
• Bijection and Inverse Function

Related Concepts¶

• Inverse Function Theorem

Sources¶

• Math I: Calculus · Lecture 4

Aliases: logarithm, logarithmic function Keywords: natural logarithm, ln, log Symbols: alpha, real numbers

Last updated: 20 October 2026 03:02