Introduction to logarithms

In both of the problems above, part (d) required you to invert the normal exponential function. Instead of going from time to amount , it asked you to go from amount to time . (This is what an inverse function does—it goes the other way—remember?)

So let’s go ahead and talk formally about an inverse exponential function. Remember that an inverse function goes backward . If $f\left(x\right)=2^x$ turns a 3 into an 8, then $f^{-1}\left(x\right)$ must turn an 8 into a 3.

So, fill in the following table (on the left) with a bunch of $x$ and $y$ values for the mysterious inverse function of $2^x$ . Pick $x$ -values that will make for easy $y$ -values. See if you can find a few $x$ -values that make $y$ be 0 or negative numbers!

On the right, fill in $x$ and $y$ values for the inverse function of ${10}^x$ .

Now, let’s see if we can get a bit of a handle on this type of function.

In some ways, it’s like a square root. $\sqrt{x}$ is the inverse of $x^2$ . When you see $\sqrt{x}$ you are really seeing a mathematical question: “What number, squared, gives me $x$ ?”

Now, we have the inverse of $2^x$ (which is quite different from $x^2$ of course). But this new function is also a question: see if you can figure out what it is. That is, try to write a question that will reliably get me from the left-hand column to the right-hand column in the first table above.

Do the same for the second table above.

Now, come up with a word problem of your own, similar to the first two in this exercise, but related to compound interest.