## Subject I. (30 points)

Solve the equation $5^{3 x+1} \cdot 49^{\frac{3 x+1}{3 x+3}}=175$.

Prof. Gheorghe Lobont, National College "Mihai Viteazul" Turda
## Subject I.

We have $x \in \mathbb{R}-\{-1\}$ and the equation becomes $\left(5 \cdot 49^{\frac{1}{3 x+3}}\right)^{3 x+1}=5^{2} \cdot 7 \Leftrightarrow 5^{3 x-1} \cdot 7^{\frac{3 x-1}{3 x+3}}=1 \Leftrightarrow(3 x-1) \cdot\left(\lg 5+\frac{\lg 7}{3 x+3}\right)=0$.

The set of solutions to the equation is $S=\left\{\frac{1}{3},-1-\frac{1}{3} \log _{5} 7\right\}$.