By the Zero Product Property, either [tex]( e^{x}- e^{ \pi } )=0[/tex] or [tex] (e^{x}- \pi )=0 [/tex]. So we will solve for x in each case. We need to take the natural log of each side in both cases since x is an exponent to base e, and the natural log has a base of e. So taking the natural log of e "undo" each other, leaving us with just x. Like this: [tex] e^{x}-e^ \pi =0 [/tex] so [tex]e^x=e^ \pi [/tex]. Taking the natural log of each side gives us [tex]ln(e^x)=ln(e^ \pi )[/tex]. Again, taking the natural log of base e undo each other, so [tex]x= \pi [/tex]. That's the first root. In the second case, [tex]e^x- \pi =0[/tex] so [tex]e^x= \pi [/tex]. Taking the natural log of both sides we get [tex]ln(e^x)=ln( \pi )[/tex]. That means that [tex]x=ln( \pi )[/tex]. Your solutions are [tex]x = \pi ,ln( \pi )[/tex]
