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:

$\log_b \left({-1}\right)$ is undefined in the real number line.

Proof

Aiming for a contradiction, suppose $\log_b \left({-1}\right) = y \in \R$.

Then:

$b^y = -1 < 0$

But from Power of Positive Real Number is Positive:

$b^y > 0$

The result follows by Proof by Contradiction.

$\blacksquare$

Sources

• 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 $17$