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The natural logarithm

The natural logarithm ln has base $e$ . It undoes the exponential function, so $\ln(e^x) = x$ and $e^{\ln a} = a$ when $a$ is positive. Equations of the form $e^{kx} = a$ are solved by taking ln of both sides and dividing by the coefficient of $x$ . The answer is rarely a whole number, so you round at the end and can check by substituting back.

\ln a = x \Leftrightarrow e^x = a

natural logarithm

e^{kx} = a \Rightarrow x = \frac{\ln a}{k}

solution

Symbols

$e$	the base
$k$	the coefficient in the exponent

Example

$e^{2x} = 5$ gives $x = \dfrac{\ln 5}{2} \approx 0.8047$ . Check: $e^{2 \cdot 0.8047} \approx 5.00$ .

Divide by the coefficient after you have taken the logarithm, not before.

Practise logarithms and exponential equations for free →

← Logarithmic equations · Logarithm rules →

Part of Mathematics R1: Logarithms and exponential equations.