To find the inverse function for [tex]\( f(x) = \ln 5x \)[/tex], follow these steps:
1. Start with the given function:
[tex]\[
f(x) = \ln 5x
\][/tex]
2. To find the inverse, we first express the function in terms of [tex]\( y \)[/tex]:
[tex]\[
y = \ln 5x
\][/tex]
3. Our goal is to solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]. First, exponentiate both sides of the equation to get rid of the natural logarithm:
[tex]\[
e^y = e^{\ln 5x}
\][/tex]
4. Recall that [tex]\( e \)[/tex] and the natural logarithm are inverse functions, so [tex]\( e^{\ln 5x} = 5x \)[/tex]. Therefore, we get:
[tex]\[
e^y = 5x
\][/tex]
5. Now, solve for [tex]\( x \)[/tex] by isolating it on one side of the equation:
[tex]\[
x = \frac{e^y}{5}
\][/tex]
6. Since [tex]\( y \)[/tex] was our original [tex]\( f(x) \)[/tex], we write the inverse function by swapping [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
f^{-1}(x) = \frac{e^x}{5}
\][/tex]
Hence, the correct inverse function for [tex]\( f(x) = \ln 5x \)[/tex] is:
[tex]\[
f^{-1}(x) = \frac{e^x}{5}
\][/tex]
So, the correct choice from the given options is:
[tex]\[
f^1(x)=\frac{e^x}{5}
\][/tex]
