1Cademy - Inverse Properties of Logarithms

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Logarithmic Function

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Inverse Properties of Logarithms

The Inverse Properties of Logarithms define the algebraic rules for composing exponential and logarithmic functions with the same base. For a valid base $a$ ( $a > 0$ , $a eq 1$ ) and $x > 0$ , the properties state that $a^{\log_a x} = x$ and $\log_a a^x = x$ . These relationships demonstrate that exponential and logarithmic functions are inverse operations that 'undo' one another. For the natural logarithm, these properties are expressed as $e^{\ln x} = x$ and $\ln e^x = x$ .