Which of the following is the inverse of $y=6^x$?
A. $y=\log _6 x$
B. $y=\log _x 6$
C. $y=\log _{\frac{1}{6}} x$
D. $y=\log _e 6 x$
Answer (1)
Switch x and y in the equation: x = 6 y .
Take the logarithm base 6 of both sides: lo g 6 x = lo g 6 ( 6 y ) .
Simplify using the logarithm property: lo g 6 ( 6 y ) = y .
The inverse function is y = lo g 6 x , so the answer is y = lo g 6 x .
Explanation
Finding the Inverse To find the inverse of the function y = 6 x , we need to switch x and y and solve for y .
Switching Variables Switching x and y gives x = 6 y .
Applying Logarithm To solve for y , we can take the logarithm base 6 of both sides: lo g 6 x = lo g 6 ( 6 y ) .
Simplifying Using the property of logarithms, lo g 6 ( 6 y ) = y . Therefore, y = lo g 6 x .
Final Answer Comparing the result with the given options, the inverse of y = 6 x is y = lo g 6 x .
Examples
Exponential functions and their inverses, logarithms, are used extensively in various fields. For example, in finance, they help calculate compound interest. If you invest an amount P at an annual interest rate r compounded n times per year, the amount A after t years is given by A = P ( 1 + n r ) n t . To find out how long it will take for your investment to reach a certain amount, you would use logarithms, the inverse of exponential functions, to solve for t .
