# Exponential and Logarithmic Functions

## Contents

#### Problems

Problem : Which investment earns more interest: \$10,000 at 6.2% interest compounded quarterly (4 times per year) for twenty years, or \$10,000 at 7.5% interest compounded continuously for 15 years?

\$10,000 at 6.2% interest compounded quarterly (4 times per year) for twenty years: A(t) = 10, 000(1 + )80 34, 229 dollars. \$10,000 at 7.5% interest compounded continuously for 15 years: A(t) = 10, 000e 1.125 30, 802 . The first investment earns more interest.

Problem : How soon will an investment double its value if it is invested at 5% compounded monthly?

In about 13.89 years the value of the investment will have doubled.

Problem : A city's population grows at the constant relative rate of 6%. If the original population is 10, 000 , how many people are there after ten years?

P(t) = 10, 000e .06t.10, 000e .60 18, 221 people.

Problem : The population of a bacteria culture exponentially grows from 2,000 to 20,000 in six hours. What is the constant relative rate of growth?

P(t) = P(0)e kt.20, 000 = 2, 000e 6k.e 6k = 10.k = .3838 . The growth rate is approximately 38.38%.

Problem : A substance decays exponentially and has a half-life of 2200 years. A specimen is found containing only 20% of its original amount of the substance. How old is the specimen?

M(2200) = M(0)e 2200k = M(0).k = - .000315067.M(t) = M(0)e -.000315067t = .2M(0).t = 5100 years. The specimen is about 5100 years old.