Unfortunately you cannot use logs to solve this.
However, it shouldn't take too long to use guess-and-check to see that
[tex]2^x+3^x = 2^2+3^2 = 4+9 = 13[/tex]
Showing that x = 2 is the solution to the equation.
You can use a graphing calculator to graph out [tex]y = 2^x+3^x[/tex] and [tex]y = 13[/tex]. The two curves cross at (2,13). The x coordinate of that intersection is what we're after. GeoGebra and Desmos are two free options you can pick from in terms of a graphing calculator.
