Logarithm to Own Base equals 1
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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 |
$\blacksquare$