MAT 150 – Class #18

Objectives

Graph and evaluate logarithmic functionsConvert equations to logarithmic and exponential formsEvaluate and apply common logarithmsEvaluate and apply natural logarithmsApply logarithmic properties

Example

Write each of the following exponential equations in logarithmic form.

Solution

a. 42 = 16

b.

c.

d.

1 17

7

18 8

5yx

4log 16 2

7

1log 1

7

8log 8 1

5log x y

Example

Write each of the following logarithmic equations in exponential form. Solution

a. log232 = 5

b. log100.00001 = –5

c. log81 = 0

52 32

510 0.00001

08 1

Example

Graph y = log5x. Solution

x = 5y y

X

Y

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

0

–2

–1

0

1

2

1/25

1/5

1

5

25

Example

Explain how the graph of each of the following functions compares with the graph of y = logx, find the domain, and graph each function.

a. y = log(x + 3)

b. y = 4 + log(x – 2)

c. 1

log3

y x

Example (cont)

a. y = log(x + 3)Solution The graph of y = log(x + 3) has the same shape as the graph of y = log x, but it is shifted 3 units to the left. Because the graph has been shifted left 3 units, thedomain is (–3, ∞).

Example (cont)

b. y = 4 + log(x – 2)SolutionThe graph of y = 4 + log(x - 2) has the same shape as the graph of y = log x, but it is shifted 2 units to the right and 4 units up. The vertical asymptote is x = 2,The domain is (2, ∞).

Example (cont)

c.

SolutionMultiplication of a function by a constant less than 1compresses the graph by a factor equal to that constant. The domain is (0, ∞).

1log

3y x

Example

Projections from 2010 to 2050 indicate that the percent of U.S. adults with diabetes (diagnosed and undiagnosed) can be modeled by p(x) = –12.975 + 11.851 ln x, where x is the number of years after 2000.a. Graph this function.b. Is the function increasing or decreasing? What does this

mean in the context of the application?c. What does this model predict the percent of U.S. adults

with diabetes will be in 2022?d. Use the graph to estimate the year in which this model

predicts the percent will reach 33%.

Example (cont)

Solutiona. Graph this function.

b. Is the function increasing or decreasing? What does this mean in the context of the application?The function is increasing, which means the percent of U.S. adults with diabetes is predicted to increase from 2010 to 2050.

y= –12.975 + 11.851 ln x

Example (cont)

c. What does this model predict the percent of U.S. adults with diabetes will be in 2022?The year 2022 is 22 years after 2000, so the percent in 2022 is estimated to be

p(22) = –12.975 + 11.851 ln 22 ≈ 23.7

d. Use the graph to estimate the year in which this model predicts the percent will reach 33%.

The percent is 33% when x ≈ 48.4, during 2049.

Example

Rewrite each of the following expressions as a sum, difference, or product of logarithms, and simplify if possible. a. b.

Solutiona. b.

= ¿ ln1− ln𝑥5

¿0−5 ln 𝑥¿−5 ln𝑥

Example

Rewrite each of the following expressions as a single logarithm.

a. log5x + 3log5y b.

Solutiona. log5x + 3log5y b.

= log5x + log5y3

= log5xy3

1log 5log

2 a b

1log 5log

2 a b

1/2 5log loga b

1/2

5log

a

b

Example (cont)

Rewrite each of the following expressions as a single logarithm.

c.

Solutionc.

ln 7 2lnx z

ln 7 2lnx z

2ln 7 lnx z

2

7ln

x

z

Assignment

Pg. 337-340 #1-11 odd#15-17 odd (Remember to be detailed on graphs.) #31-37 odd#39, 41