Which of the following is the inverse of [tex]$y=6^x$[/tex]?
A. [tex]$y=\log _6 x$[/tex]
B. [tex]$y=\log _x 6$[/tex]
C. [tex]$y=\log _{\frac{1}{6}} x$[/tex]
D. [tex]$y=\log _6 6 x$[/tex]
Answer (2)
Switch x and y in the equation: x = 6 y .
Take the logarithm base 6 of both sides: lo g 6 x = lo g 6 ( 6 y ) .
Simplify using the logarithm property: lo g 6 x = y .
The inverse function is: y = lo g 6 x .
Explanation
Finding the Inverse To find the inverse of the function y = 6 x , we need to switch x and y and solve for y .
Switching Variables Switching x and y , we get x = 6 y .
Applying Logarithm To solve for y , we take the logarithm base 6 of both sides: lo g 6 x = lo g 6 ( 6 y ) .
Simplifying Using the property of logarithms, lo g b ( b x ) = x , we get lo g 6 x = y . Therefore, the inverse function is y = lo g 6 x .
Final Answer Comparing this result with the given options, we see that the correct answer is y = lo g 6 x .
Examples
Exponential functions and their inverses, logarithmic functions, are used extensively in modeling growth and decay in various real-world scenarios. For example, the growth of a bacteria colony can be modeled using an exponential function, and the time it takes for a radioactive substance to decay to a certain level can be modeled using a logarithmic function. Understanding inverse functions helps in converting between these models and solving for unknown parameters.
The inverse of the function y = 6 x is found by switching the variables and taking the logarithm base 6. The inverse is expressed as y = lo g 6 x , which corresponds to option A. Therefore, the correct answer is A: y = lo g 6 x .
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