Which of the following is the inverse of $y=6^x$?
A. $y=\log _6 x$
B. $y=\log _x 8$
C. $y=\log _{\frac{1}{6}} x$
D. $y=\log _6 6 x$
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
Switching Variables To find the inverse of the function y = 6 x , we need to switch the roles of x and y and then solve for y . This means we start with the equation x = 6 y .
Applying Logarithm To isolate y , we take the logarithm base 6 of both sides of the equation x = 6 y . This gives us lo g 6 ( x ) = lo g 6 ( 6 y ) .
Simplifying the Equation Using the property of logarithms that lo g b ( b a ) = a , we simplify the right side of the equation to get lo g 6 ( x ) = y .
Identifying the Inverse Function Therefore, the inverse function is y = lo g 6 ( x ) . Comparing this to 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 the colony 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 required for an investment to reach a specific value involves logarithmic functions. In radioactive decay, the amount of a radioactive substance remaining after a certain time is modeled using an exponential function, and the time it takes for half of the substance to decay (half-life) is determined using a logarithmic function.
The inverse of the function y = 6 x is y = lo g 6 x . This is found by switching the variables and taking the logarithm of both sides. Therefore, the chosen option is A. y = lo g 6 x .
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