The logarithm identity rule is one of the fundamental properties of logarithms. It states that:

$$\log_b (b^x) = x$$

Here’s what this means:

• Definition: The logarithm (\log_b(y)) answers the question: “To what power must the base (b) be raised to yield (y)?”

• Identity Rule Explanation: When the number inside the logarithm is exactly the base raised to some power (i.e., (b^x)), then the logarithm simply “undoes” the exponentiation. That’s why:

  $$\log_b (b^x) = x$$

Example

If (b = 2) and (x = 3), then:

$$\log_2 (2^3) = \log_2 (8) = 3$$

This is because (2^3 = 8).

Why It Works

This rule works because logarithms and exponentiation are inverse operations. Applying a logarithm to an exponentiation with the same base effectively cancels out the operations.

In addition to this primary identity, there are related logarithm properties such as:

• (\log_b(1) = 0) because (b^0 = 1).
• (b^{\log_b(x)} = x) which is the inverse relationship.

Understanding the logarithm identity rule is essential because it helps simplify many expressions and solve equations involving logarithms.

Latex

\log_b (b^x) = x