Question

expand into a sum and/or difference of logarithms of z, or as a single logarithm without exponents. assume all variables represent positive real numbers. express numbers in exact simplest form. \\(\log_{8} z^{1/6} = \square\\)

Explanation:

Step1: Recall the power rule of logarithms

The power rule of logarithms states that \(\log_b(a^n) = n\log_b(a)\) for any positive real numbers \(a\), \(b\) (where \(b
eq1\)) and real number \(n\).
Here, we have \(\log_{8}(z^{1/6})\), so we can apply the power rule.

Step2: Apply the power rule

Using the power rule \(\log_b(a^n) = n\log_b(a)\) with \(b = 8\), \(a=z\) and \(n=\frac{1}{6}\), we get:
\(\log_{8}(z^{1/6})=\frac{1}{6}\log_{8}(z)\)

Answer:

\(\frac{1}{6}\log_{8}z\)