# Find inverse of exponential function

Do you know how I could compute the inverse function of the following exponential sentence?

$$y=\dfrac{e^x}{1+2e^x}$$

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Can you solve for $x$ in terms of $y$? – André Nicolas Dec 1 '12 at 21:43
Is it at all possible that you wanted $e^x$ rather than $e^2$ in the numerator? (If so, then my answer below would have to get changed.) – Michael Hardy Dec 1 '12 at 21:48
this is my result $F^-1(x)=ln((x/1-2x))$ – Rakisbro Dec 1 '12 at 21:49
Michael you're right, what I want is $e^x$ rather than $e^2$constant. – Rakisbro Dec 1 '12 at 21:53
For the amended question, the answer is correct. – André Nicolas Dec 1 '12 at 21:55

## 2 Answers

$$\text{exponentiate} \longrightarrow \text{multiply by 2}\longrightarrow\text{add 1} \longrightarrow \text{reciprocal}\longrightarrow\text{multiply by }e^2$$

What gets done last gets undone first.

So the inverse is: $$\text{Divide by }e^2\longrightarrow\text{reciprocal} \longrightarrow\text{subtract 1} \longrightarrow \text{divide by 2}\longrightarrow\text{take logarithm}$$

I.e. if $y = \dfrac{e^2}{1+2e^x}$ then $x = \log_e\left(\dfrac{\dfrac{1}{y/e^2} - 1}{2}\right)$. (Then simplify.)

LATER EDIT:

Since you've now said you wanted $e^x$ in the numerator, here's that: $$y=\frac{e^x}{1+2e^x} = \frac{1}{e^{-x}+2}$$ $$\text{multiply by }-1 \longrightarrow \text{exponentiate} \longrightarrow \text{add 2} \longrightarrow\text{reciprocal}$$

So the inverse is: $$\text{reciprocal} \longrightarrow\text{subtract 2}\longrightarrow\text{take logarithm} \longrightarrow\text{multiply by }-1$$

(Two of the steps are their own inverses.) So $$x = -\log_e\left(\frac1y-2\right)$$ (Then simplify.)

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The answer is correct. Thanks for your help. – Rakisbro Dec 1 '12 at 22:06
Sorry but I don't get how you go from $\frac{e^x}{1+2e^x}$ to $\frac{1}{e^{-x}+2}$ – MphLee May 20 '13 at 20:24
@MphLee : $\dfrac{e^x}{1+2e^x}$ $=\dfrac{e^x}{1+2e^x}\cdot\dfrac{e^{-x}}{e^{-x}}$. When you multiply the numerators you get $e^x\cdot e^{-x}=1$. When you multiply the denominators you get $(1+2e^x)(e^{-x})$ $=(e^{-x}) + (2e^x e^{-x})$. And $2e^x e^{-x}=2\cdot1$. – Michael Hardy May 20 '13 at 23:32

$$y=\dfrac{e^2}{1+2e^x}$$ $$1+2e^x=e^2/y$$ $$2e^x=e^2/y-1$$ $$e^x=\frac{e^2/y-1}{2}$$ $$x=\ln{\frac{e^2/y-1}{2}}$$

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