Question Video: Finding an Inverse Logarithmic Function Mathematics

If 𝑔(𝑥) is the inverse of the function 𝑓(𝑥) = 2^(𝑥 + 5), find 𝑔(𝑥).

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Video Transcript

If 𝑔 of 𝑥 is the inverse of the function 𝑓 of 𝑥, which is equal to two to the power of 𝑥 plus five, find 𝑔 of 𝑥.

We begin by recalling that the logarithmic function is the inverse of an exponential function. If 𝑓 of 𝑥 is equal to 𝑎 to the power of 𝑥, then the inverse of 𝑓 of 𝑥 is equal to log base 𝑎 of 𝑥 where 𝑎 is positive and not equal to one. In this question, we are told that 𝑓 of 𝑥 is equal to two to the power of 𝑥 plus five. This can be rewritten as 𝑦 is equal to two to the power of 𝑥 plus five. To find the inverse of any function, we exchange the variables 𝑥 and 𝑦 and then solve for 𝑦.

We begin with the equation 𝑥 is equal to two to the power of 𝑦 plus five. Using the fact that a logarithmic function is the inverse of an exponential function, this can be rewritten as 𝑦 plus five is equal to log base two of 𝑥. We can then subtract five from both sides of this equation so that 𝑦 is equal to log base two of 𝑥 minus five. We then replace 𝑦 with the inverse function 𝑔 of 𝑥.

The inverse of two the power of 𝑥 plus five is log base two of 𝑥 minus five.

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