To find the equation of the inverse of the function [tex]\( f(x) = 8 \log_3 (x - 2) + 4 \)[/tex], we need to follow a series of steps to solve the equation for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
1. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[
y = 8 \log_3 (x - 2) + 4
\][/tex]
2. Subtract 4 from both sides to isolate the logarithmic term:
[tex]\[
y - 4 = 8 \log_3 (x - 2)
\][/tex]
3. Divide both sides by 8 to further isolate the logarithm:
[tex]\[
\frac{y - 4}{8} = \log_3 (x - 2)
\][/tex]
4. Convert the logarithmic form to its equivalent exponential form:
[tex]\[
3^{\frac{y - 4}{8}} = x - 2
\][/tex]
5. Add 2 to both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = 3^{\frac{y - 4}{8}} + 2
\][/tex]
6. Replace [tex]\( x \)[/tex] with [tex]\( f^{-1}(x) \)[/tex] and [tex]\( y \)[/tex] with [tex]\( x \)[/tex]:
[tex]\[
f^{-1}(x) = 3^{\frac{x - 4}{8}} + 2
\][/tex]
So, the equation of the inverse function is:
[tex]\[
f^{-1}(x) = 3^{\frac{x - 4}{8}} + 2
\][/tex]
