Sure, let's find the exact value of the logarithmic expression [tex]\(\log_2 \sqrt[5]{16}\)[/tex] step by step.
1. Express the problem:
[tex]\[
\log_2 \sqrt[5]{16}
\][/tex]
2. Rewrite the expression inside the logarithm using fractional exponents:
[tex]\[
\sqrt[5]{16} = 16^{1/5}
\][/tex]
3. Substitute this back into the logarithmic expression:
[tex]\[
\log_2 16^{1/5}
\][/tex]
4. Apply the power rule of logarithms ([tex]\(\log_b (a^c) = c \cdot \log_b a\)[/tex]):
[tex]\[
\log_2 16^{1/5} = \frac{1}{5} \cdot \log_2 16
\][/tex]
5. Simplify [tex]\(\log_2 16\)[/tex]:
- Recognize that [tex]\(16\)[/tex] is a power of [tex]\(2\)[/tex]: [tex]\(16 = 2^4\)[/tex]
- Therefore, [tex]\(\log_2 16 = \log_2 (2^4) = 4\)[/tex], because [tex]\(\log_b (b^a) = a\)[/tex].
6. Substitute the simplified logarithm back:
[tex]\[
\log_2 16^{1/5} = \frac{1}{5} \cdot 4
\][/tex]
7. Multiply the fractions:
[tex]\[
\frac{1}{5} \cdot 4 = \frac{4}{5}
\][/tex]
So, the exact value of the logarithmic expression [tex]\(\log_2 \sqrt[5]{16}\)[/tex] is:
[tex]\[
\boxed{\frac{4}{5}}
\][/tex]
