To find \( \log h^3 \), we can use the power rule of logarithms, which states that \( \log a^b = b \cdot \log a \).
Given: \[ \log h = \frac{3}{5} \]
Using the power rule to rewrite \( \log h^3 \):
\[ \log h^3 = 3 \cdot \log h \]
Now, substitute the value of \( \log h \):
\[ \log h^3 = 3 \cdot \frac{3}{5} \]
Now, perform the multiplication:
\[ \log h^3 = \frac{9}{5} \]
Thus, the value of \( \log h^3 \) is:
\[ \boxed{\frac{9}{5}} \]