Logarithm of the Base Rule

The Logarithm of the Base Rule states that the logarithm of a base taken to itself is always equal to 1. Mathematically, this is written as:

Explanation

A logarithm answers the question: To what exponent must the base be raised to produce a given number?

• In this case, we are asking:
  “What power must we raise \( b \) to, in order to get \( b \)?”

• Since \( b^1 = b \), the answer is clearly 1.

Example

1. \( \log_2 2 = 1 \) because \( 2^1 = 2 \).
2. \( \log_5 5 = 1 \) because \( 5^1 = 5 \).
3. \( \log_{10} 10 = 1 \) because \( 10^1 = 10 \).

This rule is a direct consequence of the definition of logarithms and is useful in simplifying logarithmic expressions.

Latex

\log_b b = 1