Exponential and Logarithmic Functions

1. REASONING Use the properties of exponents to prove the Power Property of Logarithms.
2. OPEN ENDED Give an example of a quantity that grows or decays at a fixed rate. Write a real-world problem involving the rate and solve by using logarithms.
3. Writing in Math Use the information about banking on page 556 to explain how the natural base e is used in banking. Include an explanation of how to calculate the value of an account whose interest is compounded continuously, and an explanation of how to use natural logarithms to find the time at which the account will have a specified value in your answer.

Natural Logarithms
Continuously Compounded Intere
GET READY for the Lesson
Suppose a bank compounds
interest on accounts continuously,
that is, with no waiting time
between interest payments.
To develop an equation to
determine continuously
compounded interest, examine
what happens to the value A of
an account for increasingly larger
numbers of compounding
periods n. Use a principal P of $1,
an interest rate r of 100% or 1,
and time t of 1 year.
A=P(1.4)
24414…
26130.-
(yearly) (**)
(quarterly) 1(1+”
(monimy (13)
365
2.7145..
27181.
(daily)
8760
Chourly)
1(1+-10
+
ase e and Natural Logarithms In the table above, as n increases, the
=pression 1(1
1)) or (1 + 1)” approaches the irrational number
71828…. This number is referred to as the natural base, e.
exponential function with base e is called a
ty
tural base exponential function. The graph of
e’ is shown at the right. Natural base
ponential functions are used extensively in
ky-
ence to model quantities that grow and decay
ntinuously.
st calculators have an e* function for
luating natural base expressions.
(1, el
13
10.1)
o
KAMPLE Evaluate Natural Base Expressions
Ise a calculator to evaluate each expression to four decimal places.
KEYSTROKES: 2nd ſe’] 2 ENTER 7.389056099
7.3891
16.261
20