Answer:
[tex]x=\frac{3}{16}[/tex]
Step-by-step explanation:
Given : Exponential function [tex]9^{8x}=27[/tex]
We have to solve the given exponential equation.
Consider the given exponential function [tex]9^{8x}=27[/tex]
[tex]\mathrm{Convert\:}9^{8x}\mathrm{\:to\:base\:}3[/tex]
[tex]9^{8x}=\left(3^2\right)^{8x}[/tex]
Function becomes,
[tex]\left(3^2\right)^{8x}=27[/tex]
Convert 27 to base 3, we have,
[tex]\left(3^2\right)^{8x}=3^3[/tex]
Apply exponent rule, [tex]\left(a^b\right)^c=a^{bc}[/tex]
We have, [tex]3^{2\cdot \:8x}=3^3[/tex]
[tex]\mathrm{If\:}a^{f\left(x\right)}=a^{g\left(x\right)}\mathrm{,\:then\:}f\left(x\right)=g\left(x\right)[/tex]
We have,
[tex]2\cdot \:8x=3[/tex]
Simplify, we have,
[tex]16x=3[/tex]
Thus, [tex]x=\frac{3}{16}[/tex]
