To solve the expression [tex]\(\log _{15} 2^3\)[/tex] using the power property of logarithms, let's break down the steps.
1. Recognize the power property of logarithms: The power property states that [tex]\(\log_b (a^n) = n \cdot \log_b (a)\)[/tex]. This means that when you have a logarithm of a power, you can bring the exponent in front as a multiplier.
2. Identify the components of the given expression:
- The base of the logarithm is [tex]\(15\)[/tex].
- The argument of the logarithm is [tex]\(2^3\)[/tex].
3. Apply the power property:
- For the logarithmic expression [tex]\(\log _{15} 2^3\)[/tex], you can apply the power property to rewrite it.
[tex]\[
\log _{15} 2^3 = 3 \cdot \log _{15} 2
\][/tex]
Thus, using the power property, [tex]\(\log _{15} 2^3\)[/tex] can be rewritten as [tex]\(3 \cdot \log_{15} 2\)[/tex].
Given the multiple choices:
- [tex]\(\log_{15} 5\)[/tex]
- [tex]\(\log_{15} 6\)[/tex]
- [tex]\(2 \log_{15} 3\)[/tex]
- [tex]\(3 \log_{15} 2\)[/tex]
The correct answer is:
[tex]\[
3 \log_{15} 2.
\][/tex]
Therefore, [tex]\(\log_{15} 2^3 = 3 \log_{15} 2\)[/tex].
