The logarithm of 1 in any base is always 0.

Mathematically, this is written as:

$$\log_b(1) = 0$$

where ( b ) is any positive base (except 1).

Why is this true?

By definition, a logarithm answers the question: “To what exponent must the base be raised to get the given number?”

That is:

$$\log_b(x) = y \quad \text{means} \quad b^y = x$$

So, if we take ( x = 1 ):

$$\log_b(1) = y$$

which means:

$$b^y = 1$$

Since any number raised to the power of 0 equals 1 (( b^0 = 1 )), it follows that:

$$y = 0$$

Thus,

$$\log_b(1) = 0$$

for any valid base ( b > 0, b \neq 1 ).

Latex

\log_b(1) = 0