Find the inverse function for $y=12.263\ln(x)-45.381$

inverselogarithms

Find the inverse function for $y=12.263\ln(x)-45.381$

So I did $x=12.263\ln(y)-45.381$ and solved for $y$ to get:

$e^{\frac{x+45.381}{12.263}}$ But it says this answer is incorrect. Can anyone help me? Thank you so much!

The correct answer is:

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I suspect a change of basis formula is at work here and that my answer is actually equal to the "correct" answer. Can anyone clarify what's going on for me here?

Best Answer

Note that $$ e^{\frac{{x + 45.381}}{{12.263}}} = e^{\frac{{45.381}}{{12.263}}} \cdot e^{\frac{x}{{12.263}}} = e^{\frac{{45.381}}{{12.263}}} \cdot (e^{\frac{1}{{12.263}}} )^x \approx 40.473(1.085)^x . $$ Your answer is the correct one. The "correct answer" is just an approximation which, in fact, becomes worse and worse as $x$ becomes larger and larger. For $x=10000$, the relative error will already be about $40\%$.

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