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.

The curves e to the x and ln x mirrored in the line y = x, with the points (0, 1) and (1, e) on the exponential curve matching (1, 0) and (e, 1) on the logarithm.

$\ln$ 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.

Change the base $b$ and drag across the plot: every point $(x,b^x)$ has a mirror image $(b^x,x)$ on the graph of $\log_b$ , reflected in the line $y=x$ $x$ .

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 $\log_b x+\log_b y$ $lo g_{b} y$ . 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¶

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