Question
For each question below, algebraically find the function's inverse. 8. f(x) = 5^x
Answers
GPT-4o mini
To find the inverse of the function \( f(x) = 5^x \), we will follow these steps:
1. Replace \( f(x) \) with \( y \):
\[
y = 5^x
\]
2. Swap \( x \) and \( y \):
\[
x = 5^y
\]
3. To solve for \( y \), take the logarithm of both sides. We can use the base 5 logarithm, but we can also use the natural logarithm (ln) or common logarithm (log). Here, we will use logarithm base 5:
\[
\log_5(x) = y
\]
4. Therefore, we have:
\[
y = \log_5(x)
\]
5. Now we can express this in terms of the inverse function:
\[
f^{-1}(x) = \log_5(x)
\]
Thus, the inverse of the function \( f(x) = 5^x \) is:
\[
f^{-1}(x) = \log_5(x)
\]
1. Replace \( f(x) \) with \( y \):
\[
y = 5^x
\]
2. Swap \( x \) and \( y \):
\[
x = 5^y
\]
3. To solve for \( y \), take the logarithm of both sides. We can use the base 5 logarithm, but we can also use the natural logarithm (ln) or common logarithm (log). Here, we will use logarithm base 5:
\[
\log_5(x) = y
\]
4. Therefore, we have:
\[
y = \log_5(x)
\]
5. Now we can express this in terms of the inverse function:
\[
f^{-1}(x) = \log_5(x)
\]
Thus, the inverse of the function \( f(x) = 5^x \) is:
\[
f^{-1}(x) = \log_5(x)
\]