Question

Why is it important that the definition of logarithms states that the base of the logarithm does not equal 1?

Answer 1

The change of base rule says
$\log_b(x) = \frac{\log(x)}{\log(b)}$

If b = 1, then we will have

$\log_b(x) = \frac{\log(x)}{\log(b)}$

$\log_1(x) = \frac{\log(x)}{\log(1)} = \frac{\log(x)}{0}$

which is NOT possible. We cannot divide by zero. So this is why b = 1 is NOT allowed.

Answer 2

The base of any logarithm is also the base of a power function: Example

log₁₀(x) = 2 → x = 10²
log₄ (x) = 5 → x = 4⁵

If the base was 1, then all exponents on 1 would yield 1