Use the power property to rewrite logâ‚ƒ(9).

The Correct Answer and Explanation is:

Final Answer:

log⁡3(9)=log⁡3(32)=2\log_3(9) = \log_3(3^2) = 2

Explanation

To simplify log⁡3(9)\log_3(9) using the power property of logarithms, we first need to express 9 as a power of the base 3.

Step 1: Express the number as a power of the base

Notice that: 9=329 = 3^2

So, we rewrite the logarithm: log⁡3(9)=log⁡3(32)\log_3(9) = \log_3(3^2)

Step 2: Apply the power property of logarithms

The power property states that: log⁡b(an)=n⋅log⁡b(a)\log_b(a^n) = n \cdot \log_b(a)

Here, a=3a = 3, n=2n = 2, and b=3b = 3. Applying the rule: log⁡3(32)=2⋅log⁡3(3)\log_3(3^2) = 2 \cdot \log_3(3)

Step 3: Simplify using the identity log⁡b(b)=1\log_b(b) = 1

log⁡3(3)=1\log_3(3) = 1

So: 2⋅log⁡3(3)=2⋅1=22 \cdot \log_3(3) = 2 \cdot 1 = 2

Thus, log⁡3(9)=2\log_3(9) = 2

Why This Works:

Logarithms are the inverse of exponents. That means: log⁡b(a)=cmeansbc=a\log_b(a) = c \quad \text{means} \quad b^c = a

In this problem: log⁡3(9)=xmeans3x=9\log_3(9) = x \quad \text{means} \quad 3^x = 9

Since 32=93^2 = 9, then x=2x = 2. So the logarithm equals 2.

Using the power property helps us simplify logarithms by pulling exponents down as multipliers. This is especially useful when the argument inside the logarithm is a perfect power of the base.

This rule is commonly used in algebra and calculus to simplify expressions, solve exponential equations, or differentiate logarithmic functions.