 The important question in financial mathematics… How much do we need, at what rate of return, to produce the amount we want? We'll begin with the compound interest scenario, where the amount a earned on a one-time investment c, earning interest compounded at a rate i for time t, is given by formula, and we know that there are four quantities of interest. The initial amount c, how much did we invest? The interest rate i, what was our rate of return? The time t, how long do we wait? And the final amount, a, c, t, how much do we want at the end of the period? And because this is an equation, if you know any three, you can find the fourth. So Professor Jeff wants to retire in five years with $500,000 in savings. He has $250 to invest. What rate of compound interest will he need to meet his goals? So he can invest c equals $250 for t equals five years, and wants a, c, t to be $500,000. And so we can set up our equation. We'll substitute in our known values. To solve this equation, it's helpful to remember, the last thing you do is the first thing you undo. On the right-hand side, we add i to 1, raise it to the fifth power, and then multiply by $250. Since the last thing we do is multiply, then since we're multiplying on the right by $250, we begin by dividing both sides by $250. Now the right-hand side is a fifth power, and since we're raising to the fifth power, we undo by taking the fifth root. Now we're adding 1. Since we're adding 1, we subtract 1 to get i, which we can now approximate. And note that if we want to turn this into a percentage, we multiply by 100 and round appropriately. And so Professor Jeff needs to find an account paying 357.31% interest per year. Probably he should have started saving a little earlier. Fortunately, this is math, so anything is possible. So a much younger Professor Jeff finds an investment that pays 5% annually. How much should he deposit to have $500,000 after 30 years? So we have our interest rate 0.05. We have our end amount 500,000 at t equals 30, and our unknown quantity is c, so we can replace in our compound interest equation and solve, and we find that c is… So our fourth unknown could be the amount of time, so suppose we have $10,000 to invest at an annual interest rate of 8%, how long until this amount will become $20,000. So we have the amount 10,000, our interest rate 0.08, our final amount 20,000. Substituting these into our amount equation gives us… And the one thing we don't know is the time t. Since the right hand side is a product, we'll begin by dividing by 10,000. We now have an exponential equation, so we can hit both sides with a log and solve.