Let's determine the inverse function [tex]\( f^{-1}(x) \)[/tex] for the given function [tex]\( f(x) = 10^x \)[/tex].
### Step-by-Step Solution:
1. Start with the given function:
[tex]\[
f(x) = 10^x
\][/tex]
2. To find the inverse, we need to express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]. First, replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[
y = 10^x
\][/tex]
3. Now, we need to solve for [tex]\( x \)[/tex]. To do this, we take the logarithm of both sides. Since our function is exponential with base 10, we use the common logarithm (logarithm base 10):
[tex]\[
\log(y) = \log(10^x)
\][/tex]
4. Using the properties of logarithms, specifically that [tex]\( \log(a^b) = b\log(a) \)[/tex], we get:
[tex]\[
\log(y) = x \cdot \log(10)
\][/tex]
5. Since [tex]\( \log(10) = 1 \)[/tex]:
[tex]\[
\log(y) = x \cdot 1
\][/tex]
Therefore,
[tex]\[
\log(y) = x
\][/tex]
6. Now, solve for [tex]\( x \)[/tex]:
[tex]\[
x = \log(y)
\][/tex]
7. Replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] to express the inverse function:
[tex]\[
f^{-1}(x) = \log(x)
\][/tex]
Thus, the inverse function [tex]\( f^{-1}(x) \)[/tex] for the given function [tex]\( f(x) = 10^x \)[/tex] is:
[tex]\[
f^{-1}(x) = \log(x)
\][/tex]
So, the result is:
[tex]\[
f^{-1}(x) = \log_{10}(x)
\][/tex]
