If a is a positive number and , the exponential function with base a is
You know what this means when x is an integer; for example,
You also know what this mean if x is a rational number; for instance,
What does mean if x is not a rational number? That is, suppose x has a decimal expansion which is infinite and does not repeat itself, such as
What would mean?
There are ways of defining this precisely, but I'll take an intuitive approach which relies on limits. Look at
These numbers are
If you keep going in this way, the numbers will approach a limit, and that limit is :
In a similar way, you can think of as a limit of numbers which you get by using more and more of the decimal expansion of x.
There is a special number which is often used as a base for an exponential function:
It can be defined by
e is an irrational number, like ; its decimal expansion is infinite, and does not repeat. It may seem puzzling why anyone would want to use such a weird base for an exponential function, rather than an apparently nicer base such as 2 or 10. It turns out that calculus explains what is special about e; in fact, the exponential function satisfies
No other exponential --- , --- works out to be exactly the same as its derivative. This is why when mathematicians and scientists refer to as the exponential function, and why it is used more than other exponential functions in math and science.
Properties of exponentials. Let a be a positive number, .
Example. Let a be a positive number, . Simplify .
If an amount P (the principal) is invested at an annual interest rate of r% compounded k times a year, then after n years the investment is worth
Example. $1000 is invested at 6% annual interest, compounded monthly. How much is the investment worth after 10 years?
Example. How much money must be invested at 6% annual interest, compounded monthly, so that the investment is worth $2000 after 4 years?
$1574.20 is said to be the present value of $2000.
Now suppose there is an imaginary investment where the interest is compounded continuously. You might think that you'd make a lot of money on such an investment!
The interest formula is
Write this as
Letting corresponds to continuous compounding, since k is the number of times you compound each year. But as , , so
Thus, if a principal P is invested at r% annual interest compounded continuously for n years, the value of the investment is
Example. $500 is invested for 3 years at 6% interest compounded continuously. Find the value of the investment.
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