Example 11. Solve the equation

$$
\sqrt{\frac{3}{5}}\left(\frac{3}{5}\right)^{x-1}=\frac{\sqrt[4]{5^{3 x-4}}}{\sqrt{5}}
$$
Solution. Both sides of the given equation are positive. By taking the logarithm of both sides of the equation to the base 5, we obtain the equation

$$
(x-1) \log _{5}\left(\frac{3}{5}\right)+\frac{1}{2} \log _{5}\left(\frac{3}{5}\right)=\frac{3 x-4}{4}-\frac{1}{2}
$$

i.e., the equation

$$
x\left(\log _{5} 3-1\right)-\log _{5} 3+1-\frac{1}{2}+\frac{1}{2} \log _{5} 3=\frac{3}{4} x-1-\frac{1}{2},
$$

equivalent to equation (4). From this, we find

$$
x \frac{4 \log _{5} 3-7}{4}=\frac{-4+3 \log _{5} 3}{2}
$$

i.e., $x=\frac{2\left(3 \log _{5} 3-4\right)}{4 \log _{5} 3-7}$.