Equations and Powers
When solving equations involving powers equal to other powers, we must make sure the bases are the same on both sides. We can then equate the powers and write down an equation to solve.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
\[\begin{equation} \begin{aligned} & 12x+15=2x-8 \\ & 12x-2x=-8-15 \\ & 10x=-23 \\ & x= - \frac{23}{10}\end{aligned} \end{equation}\]