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Logarithmic equations

A logarithmic equation has the unknown in the exponent. You isolate the power and take the logarithm of both sides. The common logarithm and the natural logarithm give the same answer, because the ratio of two logarithms does not depend on the base. Finally, check that the solution lies in the domain, so that you never take the logarithm of a negative number or of zero.

b^x = a \Rightarrow x = \frac{\lg a}{\lg b}

unknown exponent

\lg a = x \Leftrightarrow 10^x = a

common logarithm

Symbols

$x$	the unknown exponent
$b$	the base
$a$	the power value

Example

$3 \cdot 2^x = 48$ gives $2^x = 16$ . Then $x = \dfrac{\lg 16}{\lg 2} = 4$ , since $2^4 = 16$ .

Take the logarithm only once the power stands alone on one side.

Practise logarithms and exponential equations for free →

The natural logarithm →

Part of Mathematics R1: Logarithms and exponential equations.