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Sagot :
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]
### 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]
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