Question - Finding Inverse of Exponential Function
Solution:
To find the inverse of the exponential function $$ y = 4^x $$, you need to solve for x in terms of y.Starting with the original function:\[ y = 4^x \]Swap the roles of x and y to begin finding the inverse function:\[ x = 4^y \]Now, solve for y by taking the logarithm with base 4 of both sides (since 4 is the base of the exponential function):\[ \log_4(x) = \log_4(4^y) \]By the properties of logarithms, $$ \log_b(b^a) = a $$, so:\[ \log_4(x) = y \]Therefore, the inverse function is:\[ y = \log_4(x) \]Looking at the options provided:A. $$ y = x^{-4} $$ - Incorrect, because this represents a power function, not a logarithmic function.B. $$ y = (\frac{1}{4})^x $$ - Incorrect, this is another exponential function, not the inverse of $$ 4^x $$.C. $$ y = \log_4(x) $$ - Correct, as proved above.D. $$ y = - \log_4(x) $$ - Incorrect, this is the negative of the logarithmic function.The correct answer is C: $$ y = \log_4(x) $$.
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