Problem 3

Solve the equation

x^log₂9 + x²(3^log₂x) = x^log₂15.

Solution. Let x = 2^y.So, the given equation in terms of y is

2^{log₂(9^y)} + 4^y(3^{log₂(2^y)}) = 2^{log₂(15^y)}

then

⎛⎝(3)/(5)⎞⎠^y + ⎛⎝(4)/(5)⎞⎠^y = 1.

The function from the left side is decreasing, therefore the only real solution is y = 2 or x = 1.