What is the Inverse function of $y = 10^{-x}$? Steps are appreciated.

Question

What is the inverse of $y = 10^{-x}$?

These are my steps for the problem.

Step 1 $y = 10^{-x}$.

Step 2 $x = 10^{-y}$ by inverse substitution.

Step 3 $10^y(x) = 1$.

Step 4 $10^y = \frac{1}{x}$.

Step 5 $y = \log \frac{1}{x}$.

Step 6 $y = -\log x$.

Then how is the inverse $y = 10^{-x}$? I do not understand why $- \log x$ is equal to $10^{-x}$.

Answer 1

You do not assume they are equal to be inverses. The two functions $10^{-x}$ and $-\log x$ are inverses of each other $x>0$ because $$10^{-(-\log_{10} x)} = 10^{\log_{10} x} = x >0,$$ and $$-\log_{10} (10^{-x}) = -(-x)\log_{10}(10) = x.$$

Answer 2

To get the inverse function we first switch $x$ and $y$ to get $$x = 10^{-y}$$ Solving for this by taking the $\log$ of both sides and multiplying by $-1$, we get that $$y = -\log_{10} (x)$$ This is the inverse function to $y=10^{-x}$
Note that we can check this by plugging it back in. $10^{-(-\log_{10})} = 10^{\log_{10} x} = x$ by identity. What you are confusing is that $y$ is an arbitrary variable... if you have two functions $y=f(x)$ and $y=g(x)$ that does not mean that $f(x) = g(x)$. For example, let's take $y=5x$ and $y=x^2$. These describe two independent functions... we just use the variable $y$ for both. Obviously $5x \neq x^2$

Answer 3

I think your confusion begins with step $2$. Not that the step is wrong but the way you have proceeded and the problem you are facing is due to that step only. So better leave that step out and follow the next steps accordingly.

From $$y=f(x)=10^{-x}$$ we can say that $$-x=\log_{10}y$$ or $$x=\log_{10} \frac{1}{y}$$

From above we can say that $$x=f^{-1}y$$ So $$f^{-1}y=\log_{10} \frac{1}{y}$$ or $$\color{red}{f^{-1}x=\log_{10} \frac{1}{x}}$$ where $f^{-1}x$ is the inverse function.

Hope this helps.

Answer 4

To be clear the inverse is $-\log_{10}x$, and you had to check that $y = 10^{-x}$ is one-to-one, but other than that, what did is fine. You showed that the inverse of $y= 10^{-x}$ is $y = -\log_{10} x$. In general inverses are not equal. In $\mathbb R^2$, I believe you can take the inverse to be a reflection about the line $y = x$. Read more here.

Answer 5

$$y=10^{-x}$$ $$lny=-xln10$$ $$\frac{lny}{ln10}=-x$$ $$log_{10}y=-x$$ $$x=- log_{10}y $$