Lesson 3.2

Read: Pages 207-209

3.2 Handout1-49 (ODD), 55, 59, 63, 68, 69-75 (ODD)

Logarithmic Function and Their Graphs

Objective

Students will know how to recognize, graph, and evaluate logarithmic functions.

y loga x

Common Logarithm - a logarithm with base 10.

log10 or log

Natural Logarithm - a logarithm with base e.

loge or ln

Write each of the following equations in exponential form.

log5 1 0

log3

1

9

2

logx 2 3

ln x 2

Write each of the following equations in logarithmic form.

53 125

25

3

2 1

125

100.5 10

e 5 y

e5x y 6

Simplify each of the following expressions.

log3 81

log8 8

log 10

log27 3

log5 0.2

log9 1

log7 76

Simplify each of the following expressions.

ln e5

ln1

e

ln e2x

Simplify each of the following expressions.

10log 4

e ln 2

e ln x

Find the value of x in each of the following equations. Write all answers in simplest form.

log5 125 x

logx 36 2

log32 x 1

5

log16 32 x

ln x 5

ln x 2 3

Solve for x. Give an exact answer, and then use a calculator to round that answer to the nearest thousandth.

10x 45

ex 4

7

e5x1 7

10x2 1 7

Graphs of Logarithmic Functions

Step 1:

Step 2: Use your value of h (horizontal shift) to relocate your vertical asymptote.

Use your transformations to relocate the point (1,0).

Step 3: Plot a point to the right and left of the relocated point (1,0).

Step 4: Sketch your graph.

𝑓 (𝑥 )=𝑎 log𝑏 (𝑥−h )+𝑘

(a) Use transformations to sketch the graph of the function. Clearly label any asymptotes. (b) State the domain and range of the function. (c) State whether the graph is increasing or decreasing.

f x log6(x 4) 2

f x ln x 4

f x log x 1

Algebraically find the domain of each of the following functions.

f x log5 x

f x log x 3

f x log3 3 4x 5