What is the value of log, base, 5, start root, start index, 4, end index, 5, end rootlog 5​ 4 5​ ?

Sure, let's solve the problem step by step.

We need to find the value of $\log_5 \sqrt[4]{5}$.

Step 1: Express the fourth root in exponential form. $\sqrt[4]{5} = 5^{1/4}$

Step 2: Substitute this into the logarithm. $\log_5 (5^{1/4})$

Step 3: Use the logarithm power rule, which states $\log_b (a^c) = c \log_b (a)$. $\log_5 (5^{1/4}) = \frac{1}{4} \log_5 (5)$

Step 4: Simplify using the fact that $\log_b (b) = 1$. $\log_5 (5) = 1$

Step 5: Substitute back into the equation. $\frac{1}{4} \log_5 (5) = \frac{1}{4} \times 1 = \frac{1}{4}$

So, the value of $\log_5 \sqrt[4]{5}$ is $\frac{1}{4}$.