## Equations and Powers

Example: Solve

\[3^{2x+5}=9^{4x-6}\]

We can write

\[9=3^2 \rightarrow 9^{4x-6}=3^{2(4x-6)}=3^{8x-12}\]

.The equation becomes

\[3^{2x+5}=3^{8x-12}\]

Equating the powers gives

\[\begin{equation} \begin{aligned} & 2x+5 = 8x-12 \\ & 5+12=8x-2x \\ & 17=6x \\ & x= \frac{17}{6} \end{aligned} \end{equation}\]

Example: Solve

\[27^{4x+5}=9^{x-4}\]

We can write

\[ 27=3^3, \: 9=3^2 \rightarrow 3^{3(4x+5)}=3^{12x+15}, 9^{x-4}=3^{2(x-4)}=3^{2x-8}\]

.The equation becomes

\[3^{12x+15}=3^{2x-8}\]

Equating the powers gives

\[\begin{equation} \begin{aligned} & 12x+15=2x-8 \\ & 12x-2x=-8-15 \\ & 10x=-23 \\ & x= - \frac{23}{10}\end{aligned} \end{equation}\]