Logarithms allow us to easily solve for x when it is an exponent.
xb y if and only if logb y x
To find the exponent with e, we have to take the natural log (ln).
xe y if and only if lne y x
These equations mean the same thing. One is written in exponential form and the other is written in log form. When there is no base written for logs, it is assumed the base is 10. Natural logs always have a base of e. Remember to use the “hook” to solve the log form.
4. Use a calculator to evaluate the logarithm. Round answers to three decimal places.
a. log 9 0.954
b. ln 2.17 0.775
c. log 0.101 –0.996
d. ln 150 5.011
Compound Interest Continuously Compounded Interest
Consider an initial principal P deposited into an account that pays interest at an annual rate r (expressed as a decimal), compounded n times per year. The amount A in the account after t years is given by the equation:
1nt
rA P
n
rtA Pe
When interest is compounded continuously, the amount A in an account after t years, where P is the principle and r is the interest rate, is given by the formula:
A
r
t
n
P
balance in account ($ in bank)
interest rate MUST BE IN DECIMAL FORM!!!
time in years that money is in the account
number of times per year that money is compounded (recalculated)
principal in account (starting $)
5. If $1,000 is deposited in an account at 8% annual rate compounded quarterly, find the balance after:
a) 1 yr
1nt
rA P
n
A =
P =
r =
n =
t =
?
1,000
0.08
4
1
4 10.08
1,000 14
A
41,000 1 0.2A
1,000 1.0824A$1,082.43A
5. If $1,000 is deposited in an account at 8% annual rate compounded quarterly, find the balance after:
b) 50 yrs
1nt
rA P
n
A =
P =
r =
n =
t =
?
1,000
0.08
4
25
4 500.08
1,000 14
A
2001,000 1 0.2A
1,000 52.484897A$52,484.90A
6. If $1,000 is deposited in an account at 8% annual rate compounded monthly, find the balance after:
a) 1 yr
1nt
rA P
n
A =
P =
r =
n =
t =
?
1,000
0.08
12
1
12 10.08
1,000 112
A
121,000 1 0.0067A
1,000 1.083A$1,083.00A
6. If $1,000 is deposited in an account at 8% annual rate compounded monthly, find the balance after:
b) 50 yrs
1nt
rA P
n
A =
P =
r =
n =
t =
?
1,000
0.08
12
50
12 500.08
1,000 112
A
6001,000 1 0.0067A
1,000 53.87818A$53,878.18A
7. If $1000 is deposited in an account that pays 8% annual interest compounded continuously, find the balance after 50 years.
rtA Pe
A =
P =
r =
t =
?
1,000
0.08
50
0.08 501,000A e 41,000A e
1,000 54.59815A
$54,598.15A