Practice with Logarithmic Inverse functions! For fun :)

$$f(x) = \frac{e^{2x}-1}{2-e^{2x}}$$

Question

Practice with Logarithmic Inverse functions! For fun :)

$$f(x) = \frac{e^{2x}-1}{2-e^{2x}}$$

jhannybean · Answer

Lmao.

anonymous · Answer

1/2 ln something..

give me a sec xD

anonymous · Answer

1/2 ln ((2x+1)/(x+1)) ?

jhannybean · Answer

Well, you dont even need to switch the variables first before simplifying I guess.

anonymous · Answer

i'm used to it. 
teachers and their solutions.
need to do their way if you want the mark :P

jhannybean · Answer

$$y=\frac{e^{2x}-1}{2-e^{2x}}$$$$2y-ye^{2x} = e^{2x}-1$$$$2y+1=e^{2x}+ye^{2x}$$$$e^{2x}(y+1)=2y+1$$$$e^{2x}=\frac{2y+1}{y+1}$$$$\ln e^{2x} =\ln\left(\frac{2y+1}{y+1}\right)$$$$2x=\ln\left(\frac{2y+1}{y+1}\right)$$$$f^{-1}(x) = \ln\left[\left(\frac{2x+1}{x+1}\right)^{1/2} \right]$$