Logarithm of One
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