Common & Natural Logarithms

The natural logarithm, base $e$. It is the inverse of $e^x$.

A logarithm answers 'what power?' Two bases are used so often they get their own notation: $\log$ means base $10$ , and $\ln$ means base $e$ .

Reading $\log 1000$ as 'ten to what power is 1000?' makes evaluation quick. The answer is the exponent.

Common log (base 10)

$\log x$ with no base written means base $10$ . So $\log 1000$ asks ' $10$ to what power is $1000$ ?' — the answer is $3$ .

Negative results appear for values below 1: $\log \dfrac{1}{1000} = \log 10^{-3} = -3$ .

$\ln x$ means base $e$ . So $\ln e = 1$ and $\ln 1 = 0$ , since $e^1 = e$ and $e^0 = 1$ .

Natural logs pair with continuous growth: they undo $e^x$ , which is why they appear when solving $A = Pe^{rt}$ for time.

Evaluate $\log \dfrac{1}{1000}$ .

Write as a power of 10

\dfrac{1}{1000} = 10^{-3}

The log is the exponent

-3

Answer: $-3$

Evaluate $\ln e$ .

e to what power is e?

e^1 = e

The exponent is 1

1

Answer: $1$

What base does log use?

With no base written, $\log$ means base $10$ — the common logarithm.

What does ln mean?

The natural logarithm, base $e$ . It is the inverse of $e^x$ .

How do I read a logarithm?

As 'the base to what power gives this number?' The value of the log is that power.

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