1. Graph y = b for 0 < b < 1x
EXAMPLE 1
Graph y = 1 2 x
SOLUTION
STEP 1
Make a table of values
STEP 2
Plot the points from the table.
STEP 3
Draw, from right to left, a smooth curve that begins just above the x-axis, passes through the plotted points, and moves up to the left.

2. EXAMPLE 2
Graph y = ab for 0 < b < 1 x
Graph the function.
a. Graph y = 2 1 4 x
SOLUTION
Plot (0, 2) and Then,
from right to left, draw a curve that begins just above the x-axis, passes through the two points, and moves up to the left. 1, 1 2 a.

3. EXAMPLE 2
Graph y = ab for 0 < b < 1 x
Graph the function.
b. Graph y = –3 2 5 x
SOLUTION
Plot (0, –3) and
Then,from right to left, draw a curve that begins just below the x-axis, passes through the two points,and moves down to the left. b.
1, – 6 5

4. Graph y = ab + k for 0 < b < 1 EXAMPLE 3
x – h
EXAMPLE 3
Graph y = –2. State the domain and range. 1 2
x+1
SOLUTION
Begin by sketching the graph
of y = , which passes
through (0, 3) and Then
translate the graph left 1 unit and down 2 units .Notice that the translated graph passes through (– 1, 1) and 3 1 2 x 1,
– 1 0,

5. Graph y = ab + k for 0 < b < 1 EXAMPLE 3
x – h
EXAMPLE 3
The graph’s asymptote is the line y = –2. The domain is all real numbers, and the range is y > –2.

6. EXAMPLE 4
Solve a multi-step problem
A new snowmobile costs $4200. The value of the snowmobile decreases by 10% each year.
Snowmobiles
• Write an exponential decay model giving the snowmobile’s value y (in dollars) after t years. Estimate the value after 3 years.
• Graph the model.
• Use the graph to estimate when the value of the snowmobile will be $2500.

7. Solve a multi-step problem
EXAMPLE 4
Solve a multi-step problem
SOLUTION
STEP 1
The initial amount is a = 4200 and the percent decrease is r = So, the exponential decay model is:
y = a(1 – r) t
Write exponential decay model.
= 4200(1 – 0.10) t
Substitute 4200 for a and 0.10 for r.
= 4200(0.90) t
Simplify.
When t = 3, the snowmobile’s value is y = 4200(0.90)3 = $

8. EXAMPLE 4
Solve a multi-step problem
STEP 2
The graph passes through the points (0, 4200) and (1, 3780).It has the t-axis as an asymptote. Plot a few other points. Then draw a smooth curve through the points.
STEP 3
Using the graph, you can estimate that the value of the snowmobile will be $2500 after about 5 years.

9. (5 ) ( ) EXAMPLE 1 Simplify natural base expressions
Simplify the expression. a. e2 e5 =
e2 + 5 = e7 b. 12 e4 3 e3 =
e4 – 3 4 = 4e (5 ) c.
e –3x 2 = 52 (
e –3x ) 2 = 25
e –6x = 25
e6x

10. EXAMPLE 2
Evaluate natural base expressions
Use a calculator to evaluate the expression.
Expression
Keystrokes
Display a. e4
[ ] ex 4 b.
e –0.09
[ ] ex
0.09

11. EXAMPLE 3
Graph natural base functions
Graph the function. State the domain and range. a. y =
3e 0.25x
SOLUTION
Because a = 3 is positive and r = 0.25 is positive, the function is an exponential growth function. Plot the points (0, 3) and (1, 3.85) and draw the curve.
The domain is all real numbers, and the range is y > 0.

12. EXAMPLE 3
Graph natural base functions
Graph the function. State the domain and range. b. y =
e –0.75(x – 2)
+ 1
SOLUTION
a = 1 is positive and r = –0.75 is negative, so the function is an exponential decay function. Translate the graph of y = right 2 units and up 1 unit.
e –0.75x
The domain is all real numbers, and the range is y > 1.

13. EXAMPLE 4
Solve a multi-step problem
Biology
The length l (in centimeters) of a tiger shark can be modeled by the function
e –0.178t
l = 337 – 276
where t is the shark’s age (in years).
• Graph the model.
• Use the graph to estimate the length of a tiger shark that is 3 years old.

14. EXAMPLE 4
Solve a multi-step problem
SOLUTION
STEP 1
Graph the model, as shown.
STEP 2
Use the trace feature to determine that l when t = 3.
The length of a 3year-old tiger shark is about 175 centimeters.
ANSWER

15. Model continuously compounded interest
EXAMPLE 5
Model continuously compounded interest
Finance
You deposit $4000 in an account that pays 6% annual interest compounded continuously. What is the balance after 1 year?
SOLUTION
Use the formula for continuously compounded interest. A =
Pert
Write formula. =
4000
e0.06(1)
Substitute 4000 for P, 0.06 for r, and 1 for t.
Use a calculator.
The balance at the end of 1 year is $
ANSWER

16. EXAMPLE 1
Rewrite logarithmic equations
Logarithmic Form
Exponential Form a. = 2
log 8 3 23 = 8 b. 4
log 1 = 40 = 1 = c. 12
log 1
121 = 12 4 = –1 1 = d.
1/4
log –1 4

17. EXAMPLE 2
Evaluate logarithms
Evaluate the logarithm. 4
log a. 64
SOLUTION b
log
To help you find the value of y, ask yourself what power of b gives you y.
4 to what power gives 64? a. 4
log 43
64, so = 3. 64 b. 5
log
0.2
5 to what power gives 0.2? b. =
5–1
0.2, so
–1.
0.2 5
log

18. EXAMPLE 2
Evaluate logarithms
Evaluate the logarithm. c.
1/5
log
125
SOLUTION b
log
To help you find the value of y, ask yourself what power of b gives you y. c.
to what power gives 125? 1 5 = –3 1 5
125, so
1/5
log
125
–3. d. 36
log 6
361/2
6, so 36
log 6 = 1 2 . d.
36 to what power gives 6?

19. EXAMPLE 3
Evaluate common and natural logarithms
Expression
Keystrokes
Display
Check a.
log 8 8 8 b.
ln 0.3 .3 –
0.3
e –1.204

20. EXAMPLE 4
Evaluate a logarithmic model
Tornadoes
The wind speed s (in miles per hour) near the center of a tornado can be modeled by
where d is the distance (in miles) that the tornado travels. In 1925, a tornado traveled 220 miles through three states. Estimate the wind speed near the tornado’s center.
93 log d + 65 s =

21. Evaluate a logarithmic model
EXAMPLE 4
Evaluate a logarithmic model
SOLUTION
93 log d + 65 s =
Write function.
= 93 log
Substitute 220 for d.
93(2.342) + 65
Use a calculator. =
Simplify.
The wind speed near the tornado’s center was about 283 miles per hour.
ANSWER

22. ( ) EXAMPLE 5 Use inverse properties Simplify the expression. a.
10log4 b. 5
log
25x
SOLUTION a.
10log4 = 4 b
log x
= x b. 5
log
25x = ( 52 ) x 5
log
Express 25 as a power with base 5. = 5
log
52x
Power of a power property 2x = b
log bx
= x

23. Find inverse functions
EXAMPLE 6
Find inverse functions
Find the inverse of the function. a.
y = 6 x b.
y = ln (x + 3)
SOLUTION a. 6
log
From the definition of logarithm, the inverse of
y = 6 x is
y = x. b.
y = ln (x + 3)
Write original function.
x = ln (y + 3)
Switch x and y. = ex
(y + 3)
Write in exponential form. =
ex – 3 y
Solve for y.
ANSWER
The inverse of y = ln (x + 3) is y =
ex – 3.

24. EXAMPLE 7
Graph logarithmic functions
Graph the function. a. y = 3
log x
SOLUTION
Plot several convenient points, such as (1, 0), (3, 1), and (9, 2). The y-axis is a vertical asymptote.
From left to right, draw a curve that starts just to the right of the y-axis and moves up through the plotted points, as shown below.

25. EXAMPLE 7
Graph logarithmic functions
Graph the function. b. y =
1/2
log x
SOLUTION
Plot several convenient points, such as (1, 0), (2, –1), (4, –2), and (8, –3). The y-axis is a vertical asymptote.
From left to right, draw a curve that starts just to the right of the y-axis and moves down through the plotted points, as shown below.

26. EXAMPLE 8
Translate a logarithmic graph
Graph State the domain and range. y = 2
log
(x + 3) + 1
SOLUTION
STEP 1
Sketch the graph of the parent function y = x, which passes through (1, 0), (2, 1), and (4, 2). 2
log
STEP 2
Translate the parent graph left 3 units and up 1 unit. The translated graph passes through (–2, 1), (–1, 2), and (1, 3). The graph’s asymptote is x = –3. The domain is x > –3, and the range is all real numbers.