Question

Need help with a simple problem 2^x=3^x+1

Answer 1

For $x\ \textgreater \ 0$, $2^x\ \textless \ 3^x$, all the more $2^x\ \textless \ 3^x+1$, therefore no solution.

For $x=0$, we have $2^0=3^0+1 \Rightarrow 1=1+1 \Rightarrow 1=2$. Obviously that' false.

For $x\ \textless \ 0$ both $2^x$ and $3^x$, where $2^x\ \textgreater \ 3^x$, belong to the range $(0,1)$, then $3^x+1$ belong to the range $(1,2)$.
Those ranges don't have a common part, therefore, again, no solution.

So, the equation $2^x=3^x+1$ doesn't have a solution.