Chapter 6

Exponential and Logarithmic Functions and Applications

Section 6.5

Section 6.5 Additional Exponential and Logarithmic Models

• Compound Interest and Present Value

• Continuous Compounding and Present Value

Periodic Compound Interest Formula

The accumulated amount, A, after t years in an account with principal P invested at an annual interest rate, r (expressed as a decimal value), compounded n times per year is given by

nt

nr

A

1P

Common compounding periods: semiannually (twice a year), quarterly (four times a year), monthly (12 times a year),or daily (365 times a year).

Zule will invest $10,000 and will not use any part of it for 7 years. She has been presented with the following two options:(1) Investment account paying 2.55% compounded monthly.(2) Investment account paying 5.14% compounded quarterly.

a. Which of these options would produce the highest value for the initial investment? Round to the nearest dollar.

We will use the periodic compound interest formula.For option (1), P = 10,000, r = 0.0255, n = 12, and t = 7. For option (2), P = 10,000, r = 0.0514, n = 4, and t = 7.

$11,95212

0.025510000A

712

1

$14,2984

0.051410000A

74

1

Option (2) will yield the highest value, at $14,298.

b. How long would it take Zule’s $10,000 to grow to $25,000 if she invested it at 5.72% annual interest compounded semiannually? Round to the nearest year.

We will use the periodic compound interest formula with A = 25,000, P = 10,000, r = 0.0572, n = 2, and solve for t.

years161.0286

2.5

2.5

2.5

2.5

2.52

0.0572

2

2

2

ln2ln

lnln2

lnln

t

t 1.0286

1.0286

1.0286

1

t

t

t

250002

0.057210000

2

t

1

Present Value Formula

The present value, P, of an investment that will produce a future value in an account with annual interest rate, r, compounded n times per year is given by

nt

nr

AP

1

where A = accumulated amount or future value, and t = time in years.

How much money must Zule invest today if she wants to see her money grow to $200,000 in 25 years at 4% annual interest compounded monthly? Round to the nearest dollar.

We will use the present value formula with A = 200,000, r = 0.04, n = 12, and t = 25.

If compounded monthly at 4% for 25 years, the present value for Zule’s investment would be approximately $73,698.

$73,698

2512

120.04

200000P

1

Continuous Compound Interest Formula

The accumulated amount, A, after t years in an account with principal P invested at an annual interest rate, r, compounded continuously is given by

rtPA e

Present Value Formula

The present value, P, of an investment that will produce a future value in an account with annual interest rate, r, compounded continuously is given by

where A = accumulated amount or future value, andt = time in years.

rtAP e

How much money will Zule have in her account after 7 yearsif she invests $10,000 at 5.14% compounded continuously? Round to the nearest dollar.

We will use the continuous compound interest formula with P = 10,000, r = 0.0514, and t = 7.

$14,33070.051410000A e

Determine the amount of money Zule must invest today if she wants to see her money grow to $200,000 in 25 years at 4% interest compounded continuously. Round to the nearest dollar.

We will use the present value formula with A = 200,000, r = 0.04, and t = 25.

$73,576250.04200000P e

Using your textbook, practice the problems assigned by your instructor to review the concepts from Section 6.5.