Logarithm

Prerequisites: Exponential Function. Full path (2 pages)

• Intuitive
• Concrete

Intuitive

Given how the natural log is described in math books, there’s little “natural” about it: it’s defined as the inverse of $e^x$, a strange enough exponent already.

But there’s a fresh, intuitive explanation: The natural log gives you the time needed to reach a certain level of growth. Demystifying the Natural Logarithm (ln) by Kalid Azad

Some things go up really fast, like the number of cases of coronavirus in March. Some things go down really fast, like the stock market in March.

Logarithms are a way to flatten exponential curves, so we can see and understand their structure, even when dealing with extreme/exponential growth.

Concrete

For a base $b > 0$ with $b \neq 1$ and a number $x > 0$, the logarithm $\log_b x$ is the exponent we have to put on $b$ to get $x$:

$$y = \log_b x \quad \Longleftrightarrow \quad b^y = x .$$

So $\log_{10} 1000 = 3$ and $\log_2 \tfrac{1}{8} = -3$. The logarithm is the inverse function of $x \mapsto b^x$. The most important base in physics is $e \approx 2.718$, and $\ln x \equiv \log_e x$ is the natural logarithm, the inverse of the exponential function. An equivalent definition that does not mention $e$ at all is

$$\ln x = \int_1^x \frac{dt}{t} , \qquad x > 0 .$$

Rules

All rules follow from $b^{u} b^{v} = b^{u+v}$ and $(b^u)^a = b^{au}$:

$$\begin{aligned} \log_b (xy) &= \log_b x + \log_b y , \\ \log_b (x^a) &= a \log_b x , \\ \log_b x &= \frac{\ln x}{\ln b} . \end{aligned}$$

The first rule turns multiplication into addition, which is why logarithm tables and slide rules were used for calculations. The last rule says that logarithms to different bases differ only by a constant factor.

For calculus we need

$$\frac{d}{dx} \ln x = \frac{1}{x} , \qquad \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \dots \quad (-1 < x \le 1) .$$

Worked example: half-life

A radioactive sample decays as $N(t) = N_0 e^{-\lambda t}$. The half-life $t_{1/2}$ is the time after which $N = N_0/2$. Taking $\ln$ of $e^{-\lambda t_{1/2}} = 1/2$ gives $-\lambda t_{1/2} = -\ln 2$, so

$$t_{1/2} = \frac{\ln 2}{\lambda} \approx \frac{0.693}{\lambda} .$$

The same step gives the time any exponential process $e^{\pm\lambda t}$ needs to change by a given factor $f$: $t = \ln f/\lambda$.

Where logarithms appear in physics

• Entropy: $S = k_B \ln \Omega$, where $\Omega$ is the number of microstates. The product rule makes entropy additive: two independent systems have $\Omega = \Omega_1 \Omega_2$ and therefore $S = S_1 + S_2$.
• Decibels: a power ratio $P/P_0$ corresponds to $10 \log_{10}(P/P_0)$ dB.

• Log-log plots: a power law $y = A x^n$ becomes $\log y = \log A + n \log x$, a straight line with slope $n$. This is the standard way to read off exponents from data.

• The best introduction is Demystifying the Natural Logarithm (ln) by Kalid Azad

• See also How To Think With Exponents And Logarithms by Kalid Azad and Using Logarithms in the Real World by Kalid Azad

Path to this page

1. Calculus
2. Exponential Function