General Logarithm/Examples/Base b of 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:
- $\log_b 1 = 0$
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 $17$