Notes 4­6 The Natural Base, e
Objectives:
­ Use the number e to write and graph exponential functions representing real­
world situations
­ Solve equations and problems involving e or natural logarithms
Why learn this?
Scientists use natural logarithms and carbon dating to determine the ages of ancient bones and fossils. 1
Suppose that $1 is invested at 100% interest compounded n times for one year is represented by the function As n gets very large, interest is continuously compounded. As n becomes infinitely large, the value of the function approaches a number we call e. Like π, the constant e is an irrational number. 2
Ex. Graph 3
A logarithm with a base of e is called a natural logarithm
ex.
The natural logarithmic function
exponential function is the inverse of the natural All the properties of logarithms also apply to natural logarithms
4
Ex. Simplify.
a.
b.
c. 5
Try These:
Simplify.
a.
b.
c. 6
The formula for continuously compounded interest is where A is the total amount, P is the principal, r is the annual interest rate, and t is the time in years. Ex. What is the total amount for an investment of $1000 invested at 5% for 10 years compounded continuously?
7
Try These:
Ex. What is the total amount for an investment of $100 invested at 3.5% for 8 years and compounded continuously?
8
The half­life 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.
9
Ex. A paleontologist uncovers a fossil of a saber­toothed cat in California. He analyzes the fossil and concludes that the specimen contains 15% of its original carbon­14. Carbon­14 has a half­life of 5730 years. Use carbon­14 dating to determine the age of the fossil. 10
Ex. Determine how long it will take for 650 mg of a sample of chromium­51, which has a half­life of about 28 days, to decay to 200 mg. 11