Questions: Solve the logarithmic equation. log6(6^(1/5)) The solution set is

Solution Steps

To solve the logarithmic equation \(\log_{6} \sqrt[5]{6}\), we can use the properties of logarithms and exponents. Specifically, we can express the radical as an exponent and then apply the logarithm rules to simplify.

1. Recognize that \(\sqrt[5]{6}\) can be written as \(6^{1/5}\).
2. Use the property of logarithms: \(\log_b (a^c) = c \cdot \log_b (a)\).
3. Simplify the expression.

We start with the logarithmic equation: \[ \log_{6} \sqrt[5]{6} \] We can express \(\sqrt[5]{6}\) as \(6^{1/5}\).

Using the property of logarithms, we have: \[ \log_{b} (a^c) = c \cdot \log_{b} (a) \] Thus, we can rewrite our expression as: \[ \log_{6} \sqrt[5]{6} = \frac{1}{5} \cdot \log_{6} (6) \]

Since \(\log_{6} (6) = 1\), we can simplify further: \[ \frac{1}{5} \cdot \log_{6} (6) = \frac{1}{5} \cdot 1 = \frac{1}{5} \]

Final Answer