6.1. Find all integer solutions of the equation $x \ln 27 \log _{13} e=27 \log _{13} y$. In your answer, specify the sum $x+y$ for the solution $(x, y)$, where $y$ is the smallest, exceeding 70.
Answer: 117.

Solution: The original equation is equivalent to the equation $\frac{x \ln 27}{\ln 13}=\frac{27 \ln y}{\ln 13} \Leftrightarrow x \ln 27=27 \ln y \Leftrightarrow$ $\ln 27^{x}=\ln y^{27} \Leftrightarrow 27^{x}=y^{27} \Leftrightarrow 3^{x}=y^{9}$, with $y \geqslant 1$, and thus $x \geqslant 0$. Since 3 is a prime number, either $y=1$ (then $x=0$), or $y$ is divisible by 3 and has no other prime divisors. Therefore, $y=3^{n}$, where $n \geqslant 0$. Hence, $x=9n$. Since $y>70$, then $n \geqslant 4$, the solution sought is: $(x, y)=(36,81)$. The answer is $x+y=117$.