To solve the logarithmic expression [tex]\(\log_5 \sqrt[3]{5}\)[/tex], we can follow these detailed steps:
1. Rewrite the expression in exponential form:
The cube root of 5 ([tex]\(\sqrt[3]{5}\)[/tex]) can be written as [tex]\(5^{1/3}\)[/tex]. So, the expression [tex]\(\log_5 \sqrt[3]{5}\)[/tex] can be rewritten using this:
[tex]\[
\log_5 \sqrt[3]{5} = \log_5 (5^{1/3})
\][/tex]
2. Apply the logarithmic power rule:
The power rule for logarithms states that [tex]\(\log_b (a^c) = c \cdot \log_b a\)[/tex]. Applying this rule, where [tex]\(a = 5\)[/tex], [tex]\(b = 5\)[/tex], and [tex]\(c = \frac{1}{3}\)[/tex]:
[tex]\[
\log_5 (5^{1/3}) = \frac{1}{3} \cdot \log_5 5
\][/tex]
3. Evaluate [tex]\(\log_5 5\)[/tex]:
By definition, [tex]\(\log_b b = 1\)[/tex]. So:
[tex]\[
\log_5 5 = 1
\][/tex]
4. Multiply the results:
Substituting [tex]\(\log_5 5\)[/tex] back into our expression:
[tex]\[
\frac{1}{3} \cdot \log_5 5 = \frac{1}{3} \cdot 1 = \frac{1}{3}
\][/tex]
Therefore, the value of the expression [tex]\(\log_5 \sqrt[3]{5}\)[/tex] is [tex]\(\frac{1}{3}\)[/tex] or approximately [tex]\(0.3333333333333333\)[/tex].
