MAT 213Brief Calculus

Section 1.3Exponential and Logarithmic

Functions and Models

• Last time we looked at linear functions

• What does it mean for a function to be linear?– Constant rate of change– Repeated addition of a quantity

• What about a situation where we have repeated multiplication?– Say you put $1000 into a bank account paying 5%

compounded annually• How much would you have after 1 year?

• How much would you have after 2 years?

• How much would you have after n years?

Exponential Functionsf(x) = abx

where a≠0 and b>0• a is the initial quantity (the output corresponding to

an input of zero)• b is the base or growth ( or decay) factor• How does our previous example match up?• These functions are called exponential because they

have the input variable in the exponent• We are repeatedly multiplying by b• With linear functions we have a constant change in

amount, with exponential functions we have a constant percentage change

Exponential Growth

f(x)=abx for b >1

Domain: (-∞,∞) Range: (0,∞)

Increasing and Continuous: (-∞,∞)

Horizontal Asymptote: x-axis

b is a growth factor

• Now the previous graph was for when b > 1

• For an exponential function we only need b > 0

• What happens if 0 < b < 1?

• Can b = 1?

Exponential Decay

f(x)=abx for 0 < b <1

Domain: (-∞,∞) Range: (0,∞)

Decreasing and Continuous: (-∞,∞)

Horizontal Asymptote: x-axis

b is a decay factor

Exponential Regression

Overflow!!!

Aligning the data

Let’s let 0 correspond to 1990

x f(x)

1990

1992

1994

1996

1998

2000

6.75

9.12

12.30

16.61

22.42

30.27

Doubling time and half-life• Doubling time is the amount of time it takes

for the output of an increasing exponential function to double

• Half-life is the amount of time it takes for the output of a decreasing exponential function to decrease by half

• Let’s find the doubling time for our previous example

Logarithmic FunctionsDefinition

The logarithmic function logbx is the inverse of the exponential function bx.

logby = x means bx =y

EXAMPLES

Write as a Logarithm: 52=25Write as an Exponential function: log39=2

Logarithmic FunctionsOn Your Calculator

The most frequently used basesare 10 and e

LOG calculates log10

log10y = x means 10x =y

LN calculates loge

logey = lny = x means ex = y

Below is a graph of both f(x) = ln x and f(x) = ex

What is the equation of the dashed line?

Describe the behavior of both graphs.

Logarithmic Functions

f(x)=logbx, a>1

Domain: (0,∞) Range: (-∞,∞)Increasing and Continuous: (0,∞)

Vertical Asymptote: y-axisThe graph goes through: (1,0)

Logarithmic Model

• If b > 0, we have the increasing, concave down graph we saw in the previous slide

• What will the graph look like if b < 0?

• What does a do to our model?

0where

ln)(

b

xbaxf

• #34 from book: The following table gives the average yearly consumption of peaches per person based on that person’s yearly family income when the price of peaches is $1.50 per pound.

Yearly income (tens of thousands of dollars)

Consumption of peaches (pounds per person per year)

1 5.0

2 6.4

3 7.2

4 7.8

5 8.2

6 8.6

Inverses

• As we saw earlier, exponentials and logs are inverses of each other

• Just as we saw that the data in the previous slide had a logarithmic look, if we swapped the inputs and outputs, we would have exponential looking data

• Therefore we can solve for the input of an exponential function using a log and vice versa

Composition Property of Inverses• If f and g are inverse functions, then

• This gives way to the following properties involving logs and exponentials

• Note that the log and the exponential must have the same base!

))(())((and))(())(( xfgxfgxxgfxgf

xb

xbx

xb

b

)(log

)(log

In groups let’s try the following from the book

• 9, 11, 23