To simplify the expression $\frac{1}{5} \log 4$, we can use the property of logarithms:

$a \log_b(x) = \log_b(x^a)$

In this case, we have $a = \frac{1}{5}$ and $x = 4$, so:

$\frac{1}{5} \log(4) = \log(4^{\frac{1}{5}})$

This simplifies to:

$\log(4^{\frac{1}{5}}) = \log(\sqrt[5]{4})$

Thus, $\frac{1}{5} \log 4 = \log(\sqrt[5]{4})$.

If you need the numerical value, I can calculate that for you as well. Would you like to proceed with that?

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Here are some related problems to explore:

1. Simplify $\frac{1}{3} \log 8$.
2. Express $\frac{1}{7} \log 49$ using logarithmic properties.
3. Solve $\frac{2}{5} \log 16$ in a similar manner.
4. Find the exact value of $\frac{1}{4} \log 81$.
5. Calculate $\frac{1}{n} \log x$ for a general $n$ and $x$.

Tip: Always remember logarithmic properties like $\log_b(x^a) = a \log_b(x)$, which help simplify logarithmic expressions efficiently.