Find the Solution Set of $3^x = 2^x + 1$

Question

So this is a problem, I've tried too hard to solve it, searched the internet for it, did all I could, but I had never found the solution.

I need to find the solution set of the equation with algebra only $$3^x = 2^x + 1$$

Answer 1

Hint

$$3^x = 2^x + 1 \Leftrightarrow \left(\frac{3}{2}\right)^x=1+\left(\frac{1}{2}\right)^x$$

The LHS is increasing while the RHS is decreasing.

Compare what happens with the LHS vs RHS when $x >1$ and $x<1$. [Hint compare them with $\frac{3}{2}$]