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:

$$\log_b b = 1$$

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