Find the inverse

Question

Find the inverse  $f^{-1}\left(x\right)$  of each function (on the given interval, if specified).

$f\left(x\right)=10^{-2x}$

Accepted Answer

Welcome back, everyone. Find the inverse function G inverse of X of G of X equals five to the power of negative three X were given for answer choices A states three log X divided by log five B states five log X divided by log three C negative log X divided by five log three and D negative log X divided by three log five. So first of all, let's define our function and let's replace G of X with Y because that's how Y coordinate. So Y equals five to the power of negative three X. We want to identify the inverse function meaning we have to switch X and Y so X equals five to the power of negative three Y. And since we're solving for the inverse function we want to solve or why? Since Y is part of the exponent, we're going to take the log of both sides because logarithms allow us to rewrite exponents as leading coefficients. What type of log are we going to take? Well, we're going to take the log of base 10 because those are the answer choices that use a log of base 10, we don't see any base, meaning by definition, the base is 10. So if we take a log of both sides, then log of X must be equal to log of five to the power of negative three Y. And from here, let's recall that we can bring down the exponent based on the properties of logs such that log of eight, the power of X is simply X multiplied by a log of A. So a lot of acts must be equal two. Our exponent is negative three Y and we end up with lock five. Now solving for Y, we have to divide both sides by negative log of five. So essentially Y equals the left hand side, log X divided by the right hand side. So we have negative, we log five. So we end up with a negative sign, negative sign and then we lock five in the denominator. So now if we look at the answer choices, the final answer would be negative log X divided by three log five, which is option D of the multiple choice. Thank you for watching.

Find the inverse $f^{-1}\left(x\right)$ of each function (on the given interval, if specified).
$f\left(x\right)=10^{-2x}$

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Find the inverse ﻿f−1(x)f^{-1}\left(x\right)f−1(x)﻿ of each function (on the given interval, if specified).﻿f(x)=10−2xf\left(x\right)=10^{-2x}f(x)=10−2x﻿

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Find the inverse $f^{-1}\left(x\right)$ of each function (on the given interval, if specified).
$f\left(x\right)=10^{-2x}$