# EXPONENTIAL RULES - Problem 5

Rule 3: When there are two or more exponents to the same base, multiply them.

Problem 5: You have \$1,000 in your account and will need \$3,000 in ten years. You decide to invest your money in the following manner: You invest the \$1,000 for three years at 12% compounded monthly. You then take the proceeds and invest them for one year at 11% compounded weekly. You then take the proceeds and invest them for two years at 10% compounded daily (360 days in a bank year). How much money would you have in the bank after the six years. How much interest will you need to get if you invest the proceeds for the last four years compounded annually.

Answer: At the end of six years, you will have \$1,950.47. To have \$3,000 in your bank after another four years you will need to get 11.3642% interest.

Solution:

Step 1: Calculate how much money you will have after the first three years.

Step 2: Calculate how much money you will have after the fourth year

Step 3: Calculate how much money you will have after the sixth year

Step 4: To find out how much interest you will need to get to wind up with a balance of \$3,000 after the last four years, you will have to solve the following equation

This is a very simple problem if you use logarithms. If you don't use logarithms, you will have to rely on trial and error.

Without logarithms:

Try a 12% rate as your first guess. If the rate is 12%, then

The 12% guess is too high, so try a 11% rate as your second guess. If the rate is 11%, then

The 11% guess is too low. This means the actual rate is somewhere between 11% and 12%.
Try a 11.5% rate as your third guess. If you use 11.5% as a rate, then

The 11.5% rate guess is too high. The real rate is somewhere between 11% and 11.5%.
Try a 11.25% rate as your fourth guess.

The 11.25% guess is too low. The actual rate is somewhere between 11.25% and 11.50%.
Try a rate of 11.37% as your fifth guess. If you use 11.37% as a rate, then

This is close but still a little high.
Try a rate of 11.36% as your sixth guess. If you use 11.36% as a rate, then

The 11.36% rate is close but a little low. The actual rate is between 11.36% and 11.37%.

You could keep on going in this manner until you obtained the correct answer to accuracy that you required..

OR YOU COULD USE LOGARITHMS TO FIND THE INTEREST RATE

Divide both side of the equation

by \$1,950.47 to have

Take the natural logarithm of both sides of the equation to get

By the rules of logarithms , this equation can be simplified to

or

This later expression can be written

x = .113642 or

If you want to learn more about how to use logarithms click on Logarithms

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