function inverses with exponential, why is $x = 0$?

Question

I'm trying to learn how to use function inverses to solve equations. In an exercise I know that the solution is $0$. I am unable to reach zero myself.

Solve for $x$ in the equation $g(x) = 1$, given $g(x) = e^{-2x}$.

From my book I know that the inverse of $e^x$ is $\log_e(x)$.

The solution in my textbook reads:

$$g^{-1}(x)=-\frac{1}{2}\log(x), \ x=0.$$

I'm unable to follow or understand the steps to arrive at $x = 0$?

Answer 1

$$y=e^{-2x}\implies \ln y=\ln e^{-2x}=-2x\implies x=\dfrac{\ln y}{-2}=\dfrac{-\ln y}{2}.$$

• In other words, $g(x)=e^{-2x}\implies g^{-1}(x)=\dfrac{-\ln x}{2}.$
• In the first step we have used that the inverse function of $e^x$ is $\ln x=\log_e x.$
• In the second step we just have isolated $x.$

Now, if $g(x)=1$ we get

$$x=g^{-1}(1)=\dfrac{-\ln 1}{2}=0.$$