To find the inverse function [tex]\( f^{-1}(x) \)[/tex] for the given function [tex]\( f(x) = 16^x \)[/tex], you need to reverse the roles of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the equation. Here are the steps to find the inverse function:
1. Rewrite the Function:
Start with the given function [tex]\( f(x) = 16^x \)[/tex]. For the purpose of finding the inverse, let's rewrite it using [tex]\( y \)[/tex]:
[tex]\[
y = 16^x
\][/tex]
2. Express [tex]\( x \)[/tex] in Terms of [tex]\( y \)[/tex]:
The goal is to solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]. Start by taking the logarithm of both sides of the equation. In this case, we will take the logarithm to the base 16:
[tex]\[
\log_{16}(y) = \log_{16}(16^x)
\][/tex]
Using the logarithmic property [tex]\( \log_b(a^c) = c \cdot \log_b(a) \)[/tex], we get:
[tex]\[
\log_{16}(y) = x \cdot \log_{16}(16)
\][/tex]
Since [tex]\( \log_{16}(16) = 1 \)[/tex] (because any number to the power of itself is 1 in logarithms):
[tex]\[
\log_{16}(y) = x
\][/tex]
Therefore:
[tex]\[
x = \log_{16}(y)
\][/tex]
3. Convert the Logarithmic Base:
To express the answer in a more conventional way, we can convert the base of the logarithm to a natural logarithm (base [tex]\( e \)[/tex]) or common logarithm (base [tex]\( 10 \)[/tex]). One of the properties of logarithms allows us to do so:
[tex]\[
\log_{16}(y) = \frac{\log(y)}{\log(16)}
\][/tex]
or using the natural logarithm:
[tex]\[
\log_{16}(y) = \frac{\ln(y)}{\ln(16)}
\][/tex]
4. Write the Inverse Function:
Now, express [tex]\( f^{-1}(x) \)[/tex]:
[tex]\[
f^{-1}(x) = \frac{\log(x)}{\log(16)}
\][/tex]
or using the natural logarithm:
[tex]\[
f^{-1}(x) = \frac{\ln(x)}{\ln(16)}
\][/tex]
Thus, the inverse function of [tex]\( f(x) = 16^x \)[/tex] is:
[tex]\[
f^{-1}(x) = \frac{\log(x)}{\log(16)}
\][/tex]
