Exponential and Logarithmic Functions

1
Chapter 4

• Exponential and Logarithmic Functions

2
Exponential Functions

• Definition The function f(x) defined by f(x)
  bx where b gt0 and b ? 1, and the exponent x is
  any real number, is called an exponential
  function with base b.
• Rules for exponents
• aman amn
• (am)n amn
• (ab)n anbn
• a1 a
• a0 1

3
Compound Interest

• Compound interest is an important use of
  exponential functions in business and economics
• Example
• At 5 interest, after one year 100 is worth
  100(1 0.05) 105
• After 2 years 105(1 0.05) 100(1 0.05)2
  110.25
• After 3 years 110.25(1 0.05)100(1 0.05)3
  115.76

4
Compound Interest

• In general, let P be the amount initially
  invested, let r be the interest rate, and let S
  be the value of the investment after n years.
  Then S P(1 r)n
• Correspondingly, we can ask how much 100
  received after n years is worth today P S/(1
  r)n
• Example If r 0.6, how much is 100 received 5
  years hence worth today? 74.73

5
Another Example

• Suppose that a bank offers you a Certificate of
  Deposit at 4.8 compounded quarterly for five
  years. How much is a 100 CD worth at the end of
  this period.
• With quarterly compounding, 100(1 0.012)20
  100(1.27) 127
• Without quarterly compounding, 100(1 0.048)5
  100(1.26) 126

6
Example (cont.)

• Suppose compounding monthly
• 100 (1 0.004)60 127
• Suppose compounding daily
• 100(1 0.0001)1825 127

7
The Number e

• Suppose that we invest 1 at 100 interest for
  one year. Then at the end of the year, we would
  have 1(1 1)1 2.
• Suppose now that the interest rate is still 100,
  but that there are 10 interest periods in the
  year. In this case at the end of the year we
  would have 1(1 (1/10))10 2.59
• How about 1000 interest periods 1(1
  (1/1000))1000 2.71

8
e (cont.)

• Suppose 1,000,000 interest periods 1(1
  (1/1,000,000))1,000,000 2.72
• Or, in general
• 1(1 1/n)n will provide the value of a 1
  invested at 100 interest for one year with n
  interest payment periods
• It turns out that the limit of 1(1 1/n)n as n
  goes to infinity is e 2.71828..

9
e (cont.)

• So, the interpretation here is that e
  2.71828.. Represents the amount of money we
  would have at the end of 1 year at 100 interest
  if we have continuous compounding.
• How much would continuous compounding bring over
  one year at 5 interest? e0.05 1.0513
• How much at 5 over 3 years? e0.15 1.1618
• How much at r for t years? ert

10
The Rule of 70

• e0.7 2.0138 2.00
• This means that 1 continuously compounded at 70
  interest for one year will double (reach a value
  of 2 by the end of the year)
• Or at 35 interest 1 will double in two years
• How long does it take for 1 to double at 6
  interest? A little less than 12 years.
• At 5, would be 14 years.

11
Homework

• P. 188 15,19,29,35

12
Logarithms

• Let f(x) bx. Then, the inverse function f-1(x)
  is defined as the logarithm function with base b
  and is denoted by logbx
• So, y logbx means that by x
• And, conversely, by x means that y logbx
• So, it is helpful to think of the log of a number
  as an exponent logbx is th power to which we
  must raise b to get x
• Another way to say it is, for example, log28 3
  means that 3 is the power to which we must raise
  2 to get 8 ie., 23 8.

13
Logarithms and Exponents

• 52 25 means that log525 2
• 34 81 means that log381 4
• log101000 3 means that 103 1000
• Log648 ½ means that 641/2 8
• A special logarithm logex lnx. This is
  referred to as a natural logarithm. Natural
  logarithms are used extensively in business and
  economics.

14
Solving log and exponential functions

• log2x 4, then x 24 16
• ln (x 1) 7, then loge(x 1) 7, which means
  that e7 x 1 1095.8
• e5x 4 means that ln(4) 5x or that x ln(4)/5
• Logx49 2, then x2 49, or x 7

15
Homework

• P. 195 3, 49 and maybe a few other is 1-8 and
  29-48 just to get a little more practice

16
Properties of Logarithms

• Three important properties
• logbmn logbm logbn
• Let x logbm and let y logbn
• Then bx m and by n
• This means that mn bxby b(xy)
• Or that logbmn x y logbm logb
• logbm/n logbm logbn
• Logbmr rlogbm

17
Examples

• ln (x/zw) lnx lnz lnw
• Let Q AKaLb , then lnQ lnA alnK blnL
• lnx5(x 2)8/(x 3)1/3 1/35lnx 8ln(x-2)
  ln(x 3)
• lnx - ln(x3) lnx/(x3)
• ln3 ln7 ln2 - 2ln4 ln21 ln (32)
  ln(21/32)

18
Homework

• P. 203-204, 19 (you can use a calculator if you
  like), 29, 37. maybe a few more like these just
  to make sure you are straight on the basics

19
Solving Exponential Equations

• Let 25(x 2) 5(3x 4) Solve for x
• 52(x 2) 5(3x 4)
• 2x 4 3x 4
• x 8
• Solve 5 (3)4(x 1) 12
• 4(x 1) 7/3
• (x 1)ln4 ln(7/3)
• x ln(7/3)/ln(4) 1 1.6112

20
Demand Equation

• Let p 12(1.0 0.1q)
• Then lnp (1.0 0.1q)ln12
• So, lnp/ln12 1.0 0.1q
• This means that q 10 1.0 ln(p/12)
• For example, if p 6, ln(1/2) -0.69, so q
  16.90

21
Homework

• P. 209 5,7,13,43
• Then do the blue review problems for extra
  practice ( see pp. 211-213)