Finding the inverse of a rational exponential function

Question

I am having trouble finding the inverse of the following function: $$f(x)=\frac{5e^x}{9e^{x}-5}$$

I am able to get a fair ways through the problem and through the use of the rules of logarithms have reached this point: $$\ln{y}=5x-\ln{(9e^{x} -5)}$$

I know that I need to get that last bit simplified, but I am not sure how to do it. If it were simply $\ln{9e^x}$, the problem would be trivial.

Any help as to how I can proceed is greatly appreciated.

Can you find the inverse of $g(z) = \dfrac{5z}{9z-5}$? – Daniel Fischer Sep 08 '13 at 15:44 — Daniel Fischer, Sep 08 '13 at 15:44

lab bhattacharjee · Accepted Answer · 2013-09-08T15:57:45.757

1

$$y=f(x)=\frac{5e^x}{9e^x-5}$$

$$\implies e^x(9y)-5y=5e^x\implies e^x(9y-5)=5y\implies e^x=\frac{5y}{9y-5}$$

Applying logarithm wrt $x,$ $$x=\ln\left(\frac{5y}{9y-5}\right)$$

Now, we can apply $\displaystyle\ln\left( \frac {a_1a_2\cdots}{b_1b_\cdots}\right)=\sum{\ln a_i}-\sum{\ln b_i}$

Clearly, for real $x,$ $\displaystyle\frac{5y}{9y-5}>0\iff y(9y-5)>0$

edited Sep 08 '13 at 15:57

answered Sep 08 '13 at 15:45

lab bhattacharjee

274,582

Looks to me like you swapped $e^x$ and $y$ going from the first line to the second. – walcher Sep 08 '13 at 15:50
@walcher, yes, edited for better clarity – lab bhattacharjee Sep 08 '13 at 15:58

Finding the inverse of a rational exponential function

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