If b is any positive real number with and x is any real number, is defined as follows:
Use this definition and the definition of logarithm to prove that for all positive real numbers u and b, with
Use this definition and the definition of logarithm to prove that for all positive real numbers , with .
Let be a positive real number, and be a real number with .
Define the value of as follows.
The objective is to prove .
By the definition of a logarithm,
Multiply both sides of equation , and dividing by obtained as
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