Question

\\\log_{b^m} f(x) = \frac{1}{m} \log_b f(x)\\

Explanation:

🆕 New Concept Discovered: Logarithmic Base Change Rule
How to change or scale the base of a logarithm.

Step 1: Apply the change of base formula

To prove or derive the given identity, we use the standard change of base formula for logarithms:

$$ \log_{A} C = \frac{\log_{B} C}{\log_{B} A} $$

We will apply this formula to the left side of the equation, \( \log_{b^m} f(x) \), by changing its base to \( b \).

Step 2: Rewrite the expression

Let \( A = b^m \), \( B = b \), and \( C = f(x) \). Substituting these into the change of base formula gives:

$$ \log_{b^m} f(x) = \frac{\log_{b} f(x)}{\log_{b} (b^m)} $$

Step 3: Simplify the denominator

Using the logarithm property \( \log_{b} (b^m) = m \), we can simplify the denominator:

$$ \log_{b^m} f(x) = \frac{\log_{b} f(x)}{m} $$

Step 4: Factor out the constant

We can rewrite the division by \( m \) as multiplication by \( \frac{1}{m} \):

$$ \log_{b^m} f(x) = \frac{1}{m} \log_{b} f(x) $$

This matches the right side of the given equation.

Answer:

The identity is proven by applying the change of base formula to the left-hand side:

$$ \log_{b^m} f(x) = \frac{\log_{b} f(x)}{\log_{b} (b^m)} = \frac{1}{m} \log_{b} f(x) $$