Logarithm Identity Rule
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
\[ \log_2 (2^3) = \log_2 (8) = 3 \]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