Logarithm

Logarithm

A logarithm asks for the exponent. Given $b^n=x$, the logarithm ${\log }_b(x)$ is the exponent $n$ such that $b^n=x$.

Over the real numbers, the standard assumptions are

$$b>0,b\ne 1,x>0.$$

Identities

${\log }_b(xy)={\log }_b(x)+{\log }_b(y)$ ${\log }_b\left(\frac{x}{y}\right)={\log }_b(x)-{\log }_b(y)$ ${\log }_b(x^n)=n{\log }_b(x)$ ${\log }_b(1)=0$ ${\log }_b(b)=1$ ${\log }_b(x)=\frac{{\log }_a(x)}{{\log }_a(b)}$ $b^{{\log }_b(x)}=x$ ${\log }_b(b^x)=x$

Common bases

• ${\log }_{10}(x)$: common logarithm, useful for decimal orders of magnitude.
• $\ln (x)={\log }_e(x)$: natural logarithm, central in calculus.
• ${\log }_2(x)$: binary logarithm, common in information theory and algorithms.

Natural logarithm

The natural logarithm uses Euler’s number $e$ as its base. It is special because

$$\frac{d}{dx}\ln x=\frac{1}{x},\int \frac{1}{x}dx=\ln |x|+C.$$

It also solves exponential equations cleanly:

$$A{e}^{kt}=B\Rightarrow t=\frac{1}{k}\ln \left(\frac{B}{A}\right).$$

See Growth and decay.

Why logs are useful

Logarithms turn multiplication into addition and powers into products. That makes them useful for simplifying products, measuring ratios, linearising exponential data, and handling values spread across many orders of magnitude. In physics and astronomy, logarithmic scales appear naturally in magnitude-style measures; see Cosmology distance methods.

Calculus

For base $b$,

$$\frac{d}{dx}{\log }_b(x)=\frac{1}{x\ln b}.$$

The chain rule gives

$$\frac{d}{dx}\ln (g(x))=\frac{g^{\prime }(x)}{g(x)}.$$

This pattern is often the fastest way to spot logarithmic antiderivatives.

Common pitfalls

• Taking logs of dimensional quantities without forming a dimensionless ratio first.
• Splitting sums: $\log (x+y)\ne \log x+\log y$.
• Forgetting the base in numerical work; $\ln x$ and ${\log }_{10}x$ differ by a factor of $\ln 10$.

Log-linear trick

If data follows $y=A{e}^{kx}$, then $\ln y=\ln A+kx$. Plotting $\ln y$ against $x$ should give a straight line. This is a quick diagnostic for exponential behaviour, but only after checking units and noise.

See also Exponentiation, Root, and Differentiation rules.