Which of the following is the inverse of [tex]$y=12^x$[/tex]?
A. [tex]$y=\log _{\frac{1}{12}} x$[/tex]
B. [tex]$y=\log _{12} \frac{1}{x}$[/tex]
C. [tex]$y=\log _x 12$[/tex]
D. [tex]$y=\log _{12} x$[/tex]
Answer (1)
Switch x and y in the equation: x = 1 2 y .
Take the logarithm base 12 of both sides: lo g 12 ( x ) = lo g 12 ( 1 2 y ) .
Simplify using logarithm properties: lo g 12 ( x ) = y .
The inverse function is: y = lo g 12 x .
Explanation
Finding the Inverse To find the inverse of the function y = 1 2 x , we need to switch x and y and solve for y .
Switching Variables Switching x and y , we get x = 1 2 y .
Applying Logarithm To solve for y , we take the logarithm base 12 of both sides: lo g 12 ( x ) = lo g 12 ( 1 2 y ) .
Simplifying Using the property of logarithms, we have lo g 12 ( x ) = y .
Final Answer Therefore, the inverse function is y = lo g 12 ( x ) . Comparing this with the given options, we find that the correct answer is y = lo g 12 x .
Examples
Exponential functions and their inverses, logarithmic functions, are used to model many real-world phenomena, such as population growth and radioactive decay. For example, if you know how much a population grows each year (e.g., 12 times), you can use the exponential function y = 1 2 x to predict the population after x years. Conversely, if you want to know how many years it will take for the population to reach a certain size, you would use the inverse function, y = lo g 12 x . This is also applicable in finance for calculating investment growth and decay.
