Logarithm to Own Base equals 1

Theorem

Let $b \in \R_{>0}$ be a strictly positive real number such that $b \ne 1$.

Let $\log_b$ denote the logarithm to base $b$.

Then:

$\log_b b = 1$

Proof

By definition of logarithm:

\(\ds y\)

\(=\)

\(\ds \log_b b\)

\(\ds \leadstoandfrom \ \ \)

\(\ds b^y\)

\(=\)

\(\ds b\)

Definition of Real General Logarithm

\(\ds \leadstoandfrom \ \ \)

\(\ds y\)

\(=\)

\(\ds 1\)

Definition of Power to Real Number

$\blacksquare$