# 6-4 e and Natural Logarithms

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```					6-4 e and Natural Logarithms

We have done examples already that use compound interest in the section on Exponential Models.

COMPOUND INTEREST  annually
A = P(1 + r)t  *ONLY WORKS WHEN CALCULATING YEARLY \$ ACCRUED

MORE GENERAL EQUATION
r
A = P(1 + )nt  P = starting value, r = interest rate, t = time in years,
n
n = # of times compounded in a year

Consider P = \$1 and r = 100%
Compounded annually  A = \$2
Compounded daily  A = \$2.71457
Compounded hourly  A = \$2.71813
Compounded every second  A = \$2.71828
*the amount is getting closer and closer to the irrational number “e”

e = 2.7182818245…               USED FOR CONTINUOUS CHANGE

INTEREST COMPOUNDED CONTINUOUSLY – use “pert”

A = P(ert)     P, r, and t same as above, “e” is the number/button on your calculator

EX 1. If \$10,000 is put into bonds that pays 7.35% interest compounded continuously, find the value of
the investment after one year.

10,000(e.0735) = \$10,762.69

EX 2. Suppose \$10,000 is put into a 5-year certificate of deposit that pays 7.35% interest compounded
continuously.

a. What is the balance at the end of this period?

10,000(e.0735*5) = \$14,441.20

b. How does this compare with the balance if the interest were compounded annually?

10,000(1 + .0735)5 = \$14,256.41
14,441.20 – 14,256.41 = 184.79 dollars more if you compound continuously
EX 3. Suppose \$2000 is invested in an account paying 6% annual interest for 5 years.

a. If the interest is compounded daily, how much is the investment worth after ONE year?

.06 365
2000(1 +        ) = \$2,123.66
365

b. If the interest is compounded continuously, how much is the investment worth after one year?

2000(e.06) = \$2,123.67

c. Compounded monthly, how much is it worth at the end of the 5 year period?

.06 12*5
2000(1 +       )     = \$2,697.70
12

d. Compounded continuously, how much is it worth at the end of the 5 year period?

2000(e.06*5) = \$2,699.71

EX 4. Evaluate without a calculator:

a. e0      =1           b. ln 1 = 0           c. ln e = 1         d. ln e5 = 5

CALCULATOR EXAMPLES:

ln x = 4               loge x = 4            B.O.M.
e4 = x = 54.5982

ex = 22         B.O.M
bo m            loge 22 = x           ln 22 = x = 3.0910

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