Find the best answers to your questions at Westonci.ca, where experts and enthusiasts provide accurate, reliable information. Discover reliable solutions to your questions from a wide network of experts on our comprehensive Q&A platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
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]
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.