# Problem 2. (2 points)

Vasya came up with a new operation on the set of positive numbers: $a \star b=a^{\ln b}$. Find the logarithm of the number $\frac{(a b) \star(a b)}{(a \star a)(b \star b)}$ to the base $a \star b$.
Answer: 2

Solution: Note that $a \star b=a^{\ln b}=e^{\ln a \ln b}$. Also, $a b=e^{\ln a+\ln b}$, which means that under the operation $\star$, the logarithms of the numbers are multiplied, while under multiplication of the numbers, the logarithms are added, and under division, they are subtracted.

Let $x=\ln a$ and $y=\ln b$. Then, according to what is written above, $\ln \frac{(a b) \star(a b)}{(a \star a)(b \star b)}=(x+y)(x+y)-x^{2}-y^{2}=2 x y$, and $\ln (a \star b)=x y$. The desired number is the quotient of these logarithms, which is 2.