Which of the following is the inverse of $y=12^x$?
A. $y=\log \frac{1}{12} x$
B. $y=\log _{12} \frac{1}{x}$
C. $y=\log _x 12$
Answer (1)
Switch x and y in the equation y = 1 2 x to get x = 1 2 y .
Take the logarithm base 12 of both sides: lo g 12 ( x ) = lo g 12 ( 1 2 y ) .
Simplify using the logarithm property: lo g 12 ( x ) = y .
The inverse function is y = lo g 12 ( x ) , which is not among the given options. Therefore, none of the options are correct. y = lo g 12 ( x )
Explanation
Finding the Inverse To find the inverse of the function y = 1 2 x , we need to switch the roles of x and y and solve for y .
Switching Variables Switching x and y gives x = 1 2 y .
Applying Logarithm To solve for y , we can take the logarithm base 12 of both sides: lo g 12 ( x ) = lo g 12 ( 1 2 y ) .
Solving for y Using the property of logarithms, lo g a ( a b ) = b , we get lo g 12 ( x ) = y . Therefore, the inverse function is y = lo g 12 ( x ) .
Finding the Correct Option The inverse of the function y = 1 2 x is y = lo g 12 ( x ) . Comparing this to the options given, we see that none of the options exactly match. However, we can rewrite y = lo g 12 ( x ) as y = lo g 12 ( x ) . The closest option to this is y = lo g 12 x 1 which is not the same. Let's analyze the given options:
y = lo g 12 1 x = lo g ( x ) − lo g ( 12 ) . This is not the inverse.
y = lo g 12 x 1 = lo g 12 ( x − 1 ) = − lo g 12 ( x ) . This is not the inverse.
y = lo g x 12 . This is not the inverse.
None of the given options are the correct inverse. However, if we assume that there was a typo and the question meant to ask which of the following is equivalent to lo g 12 ( x ) , then none of the options are correct.
Examples
Exponential functions and their inverses, logarithmic functions, are used extensively in modeling growth and decay in various real-world phenomena. For instance, the growth of a population can be modeled using an exponential function, and the time it takes for the population to reach a certain size can be determined using the inverse logarithmic function. Similarly, radioactive decay is modeled using exponential functions, and the half-life of a radioactive substance is calculated using logarithmic functions. These concepts are also crucial in finance for calculating compound interest and in physics for understanding phenomena like the attenuation of light.
