Which of the following is the inverse of $y=3^x$?
A. $y=\frac{1}{3^x}$
B. $y=\log _3 x$
C. $y=\left(\frac{1}{3}\right)^x$
D. $y=\log _{\frac{1}{3}} x$
Answer (2)
Switch x and y in the equation y = 3 x to get x = 3 y .
Take the logarithm base 3 of both sides: lo g 3 x = lo g 3 ( 3 y ) .
Simplify using the logarithm property: lo g 3 ( 3 y ) = y .
The inverse function is y = lo g 3 x , so the answer 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 can take the logarithm base 3 of both sides: lo g 3 x = lo g 3 ( 3 y ) .
Simplifying Using the property of logarithms, lo g 3 ( 3 y ) = y . Therefore, y = lo g 3 x .
Final Answer Comparing the result with the given options, 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 population growth, radioactive decay, and compound interest. For example, if you invest money in an account that compounds interest, you can use logarithms to determine how long it will take for your investment to reach a certain value. Similarly, in epidemiology, exponential functions are used to model the spread of infectious diseases, and logarithms can help determine the time it takes for the number of infected individuals to double.
The inverse of the function y = 3 x is y = lo g 3 x . This can be found by switching the variables and applying logarithms. Therefore, the correct answer is option B.
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