Logarithms Explained for Beginners

A logarithm answers the question: what power do I raise the base to, to get this number? Since $10^3 = 1000$, we write $\log_{10} 1000 = 3$. Logarithms undo exponents, turn multiplication into addition, and let you solve equations where the unknown is in the power.

The definition

The natural log $\ln x$ uses base $e \approx 2.71828$. On most calculators, "log" means base 10.

The three log laws

Example: $\log_2 8 + \log_2 4 = \log_2 32 = 5$. Check: $3 + 2 = 5$ ✓.

Change of base

Calculators usually only have $\log_{10}$ and $\ln$. For any other base:

Example: $\log_3 50 = \frac{\ln 50}{\ln 3} \approx \frac{3.912}{1.0986} \approx 3.561$. Check: $3^{3.561} \approx 50$ ✓.

Solving exponential equations

Solve $2^x = 1024$. Take logs of both sides and use the power law:

since $2^{10} = 1024$.

Solve $5^x = 40$:

Where logs appear

• Growth and decay: how long until a population growing 5% a year doubles? Solve $(1.05)^t = 2$ to get $t = \frac{\ln 2}{\ln 1.05} \approx 14.2$ years.
• Scales: decibels, the Richter scale and pH are all logarithmic.
• Computing: $\log_2 n$ is how many times you can halve $n$.

Common mistakes

• $\log(x + y) \ne \log x + \log y$.
• $\log x \cdot \log y$ is not $\log(xy)$.
• Using $\log$ and $\ln$ interchangeably. They give different numbers.

Practice questions

1. Evaluate $\log_5 125$.
2. Write $\log 6 + \log 5$ as a single log.
3. Solve $3^x = 20$ to 3 significant figures.

Answers: 1) $3$ 2) $\log 30$ 3) $x \approx 2.73$

Try the logarithm calculator in any base, or go the other way with the exponents calculator. Logs also give doubling times and half-lives in exponential growth and decay.

Frequently asked questions

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