Which of the following is the inverse of [tex]$y=3^x$[/tex]?
A. [tex]$y=\frac{1}{3^x}$[/tex]
B. [tex]$y=\log _3 x$[/tex]
C. [tex]$y=\left(\frac{1}{3}\right)^x$[/tex]
D. [tex]$y=\log _{\frac{1}{3}} x$[/tex]
Answer (1)
Switch x and y in the equation: x = 3 y .
Take the logarithm base 3 of both sides: lo g 3 x = lo g 3 ( 3 y ) .
Simplify using logarithm properties: lo g 3 x = y .
The inverse function is: y = lo g 3 x .
Explanation
Finding the Inverse To find the inverse of the function y = 3 x , we need to switch x and y and solve for y .
Switching Variables Switching x and y gives x = 3 y .
Applying Logarithm To solve for y , we take the logarithm base 3 of both sides: lo g 3 x = lo g 3 ( 3 y ) .
Solving for y Using the property of logarithms, lo g 3 ( 3 y ) = y . Therefore, y = lo g 3 x .
Identifying the Correct Option Comparing the result with the given options, we find that the inverse of y = 3 x is y = lo g 3 x .
Examples
Exponential functions and their inverses, logarithmic functions, are used extensively in modeling growth and decay processes. For example, 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, in finance, compound interest calculations involve exponential functions, and determining the time it takes for an investment to reach a certain value involves logarithmic functions. These concepts are also crucial in understanding radioactive decay and carbon dating in science.
