Question

the proof for the product property of logarithms requires simplifying the expression \\(\log_{b}(b^{x+y})\\) to \\(x + y\\). which property is used to justify this step?

• \\(b^{x} \cdot b^{y} = b^{x+y}\\)
• substitution
• \\(\log_{b}(b^{c}) = c\\)
• commutative property

Explanation:

⚡ Using what you learned: properties of logarithms

Step 1: Analyze the simplification step

The problem asks for the property used to simplify the expression:

$$ \log_b(b^{x+y}) \text{ to } x + y $$

Step 2: Identify the matching logarithmic property

The general identity for a logarithm with base \( b \) of an exponential expression with the same base \( b \) is:

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

Applying this identity where the exponent is \( c = x + y \):

$$ \log_b(b^{x+y}) = x + y $$

This matches the third option.

Answer:

\( \log_b(b^c) = c \)