1. Section 6: The Natural Base, e

2. U se the number e to write and graph exponential functions representing real-world situations. S olve equations and problems involving e or natural logarithms.

3. n atural logarithm n atural logarithmic function

4. S cientists use natural logarithms and carbon dating to determine the ages of ancient bones and fossils. ( See Example 4.)

5. R ecall the compound interest formula A = P (1 + n/r) nt, where A is the total amount, P is the principal, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the time in years. S uppose that $1 is invested at 100% interest (r = 1) compounded n times for one year as represented by the function f (n) = (1 + n/1 ) n. A s n gets very large, interest is continuously compounded. Examine the graph of f (n) = (1 + _n1_ )n. T he function has a horizontal asymptote. As n becomes infinitely large, the value of the function approaches approximately 2.7182818.... This number is called e. Like π, the constant e is an irrational number.

6. E xponential functions with e as a base have the same properties as the functions you have studied. The graph of f (x) = e x is like other graphs of exponential functions, such as f (x) = 3 x. The domain of f (x) = e x is all real numbers ( R ). The range is

8. A logarithm with a base of e is called a natural logarithm and is abbreviated as “ln” (rather than as log e ). Natural logarithms have the same properties as log base 10 and logarithms with other bases. T he natural logarithmic function f (x) = ln x is the inverse of the natural exponential function f(x) = e x. T he domain of f (x) = ln x is T he range of f (x) = ln x is all real numbers. A ll of the properties of logarithms from Lesson 7-4 also apply to natural logarithms.

10. T he formula for continuously compounded interest is A = Pe rt, where A is the total amount, P is the principal, r is the annual interest rate, and t is the time in years.

12. T he half-life of a substance is the time it takes for half of the substance to breakdown or convert to another substance during the process of decay. N atural decay is modeled by the function below.

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