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 (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 ) .
Solving for y 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 to model many real-world phenomena. For example, the growth of a population can be modeled by an exponential function, while the time it takes for a radioactive substance to decay can be modeled by a logarithmic function. Understanding inverse functions helps in solving problems related to growth rates and decay times. Another example is in finance, where compound interest can be modeled exponentially, and finding the time it takes to reach a certain investment goal involves using logarithms.
The inverse of the function y = 3 x is found by switching x and y , applying the logarithm base 3, and simplifying to obtain y = lo g 3 x . Therefore, the answer is option B: y = lo g 3 x .
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