7.056. $7^{\lg x}-5^{\lg x+1}=3 \cdot 5^{\lg x-1}-13 \cdot 7^{\lg x-1}$.
## Solution.

Domain: $x>0$.

From the condition

$$
7^{\lg x}-5 \cdot 5^{\lg x}=\frac{3}{5} \cdot 5^{\lg x}-\frac{13}{7} \cdot 7^{\lg x}, \quad 35 \cdot 7^{\lg x}+65 \cdot 7^{\lg x}=21 \cdot 5^{\lg x}+175 \cdot 5^{\lg x},
$$

$100 \cdot 7^{\lg x}=196 \cdot 5^{\lg x},\left(\frac{7}{5}\right)^{\lg x}=\left(\frac{7}{5}\right)^{2}$,

from which $\lg x=2$ and $x=100$.

Answer: 100.