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Logarithm as exponent

A logarithm is the exponent of an exponential function.

 

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Explanation

The logarithm is the inverse of the exponential function. It is defined for b > 0, and satisfies

b = e^ln b

Because b^x works according to the rules for logarithms and exponents, the following should apply

for each real number x. This definition, which contains powers with logarithms, is common for complex numbers.

 

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Example 1

From the definition of the logarithm follows that you can write each number as an exponential function, so

1 = e^ln (1)

And because ln (1) = 0 you get

1 = e^ln (1) = e⁰ = 1

The zeroth power always delivers the value 1.

 

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