General Logarithm/Examples/Base b of 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:

$\log_b 1 = 0$

Proof

By definition of logarithm:

\(\ds \log_b 1\)

\(=\)

\(\ds \frac {\log_e 1} {\log_e b}\)

Change of Base of Logarithm

\(\ds \)

\(=\)

\(\ds \frac 0 {\log_e b}\)

Natural Logarithm of 1 is 0

\(\ds \)

\(=\)

\(\ds 0\)

whatever $\log_e b$ happens to be

$\blacksquare$

Sources

• 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $0$
• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.2$: Numbers, Powers, and Logarithms: Exercise $17$