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Logarithm rules

The logarithm turns a product into a sum and a quotient into a difference. An exponent can be moved down as a factor in front of the logarithm. The same rules hold for both lg and ln. When you solve an equation with a sum of logarithms, you combine them into one logarithm before you remove it. Afterwards you must reject solutions that make the argument negative or zero.

\ln(ab) = \ln a + \ln b

product rule

\ln(a^n) = n\ln a

power rule

\ln 1 = 0

logarithm of 1

Symbols

$a,\ b$	positive numbers
$n$	the exponent

Example

$\ln(e^3) + \ln 1 = 3 + 0 = 3$ . The equation $\ln x + \ln(x - 1) = \ln 6$ gives $x(x - 1) = 6$ , and $3 \cdot 2 = 6$ , so $x = 3$ .

A solution that gives a negative argument, for example

x = -2

, must be rejected.

Practise logarithms and exponential equations for free →

← The natural logarithm · The chain rule →

Part of Mathematics R1: Logarithms and exponential equations.