1. Evaluate logarithms and Graph Logarithmic Functions Lesson 4.4
Honors Algebra 2
Evaluate logarithms and Graph Logarithmic Functions
Lesson 4.4

2. Goals Goal Rubric Level 1 – Know the goals.
Rewrite logarithmic equations as exponential equations.
Evaluate logarithms.
Evaluate common and natural logarithms.
Use inverse properties of logarithms and exponentials.
Find inverse functions.
Graph logarithmic functions.
Translate logarithmic functions.
Level 1 – Know the goals.
Level 2 – Fully understand the goals.
Level 3 – Use the goals to solve simple problems.
Level 4 – Use the goals to solve more advanced problems.
Level 5 – Adapts and applies the goals to different and more complex problems.

3. Vocabulary Logarithm of y with base b Common logarithm
Natural logarithm

4. Logarithm
How many times would you have to double $1 before you had $8?
You could use an exponential equation to model this situation. 1(2x) = 8.
You may be able to solve this equation by using mental math if you know 23 = 8.
So you would have to double the dollar 3 times to have $8.

5. Logarithm
How many times would you have to double $1 before you had $512?
You could solve this problem if you could solve 2x = 8 by using an inverse operation that undoes raising a base to an exponent equation to model this situation.
This operation is called finding the logarithm.
A logarithm is the exponent to which a specified base is raised to obtain a given value.

6. Logarithm
You can write an exponential equation as a logarithmic equation and vice versa.
Read logb a= x, as “the log base b of a is x.” Notice that the log is the exponent.
Reading Math

7. Logarithm

8. Logarithm
Every logarithmic equation has an equivalent exponential form:
y = loga x is equivalent to x = a y
A logarithm is an exponent!
A logarithmic function/operation is the inverse function/operation of an exponential function/operation.

9. 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

10. Your Turn:
for Example 1
Rewrite the equation in exponential form.
Logarithmic Form
Exponential Form 1. = 3
log 81 4 34 = 81 2. 7
log = 1 71 = 7 = 3. 14
log 1
140 = 1 32 = –5 1 2 = 4.
1/2
log –5 32

11. EXAMPLE 2
Evaluate logarithms
Evaluate the logarithm. 4
log a. 64 b. 5
log
0.2 b
log
SOLUTION
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
5 to what power gives 0.2? b. =
5–1
0.2, so
–1.
0.2 5
log

12. EXAMPLE 2
Evaluate logarithms
Evaluate the logarithm. c.
1/5
log
125 d. 36
log 6 b
log
SOLUTION
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.
361/2
6, so 36
log 6 = 1 2 . d.
36 to what power gives 6?

13. Your Turn:
for Example 2
Evaluate the logarithm. Use a calculator if necessary. 2
log 5. 32 25
32, so = 5. 2
log 32
ANSWER 27
log 6. 3 1 3
271/3
3, so 27
log = .
ANSWER

14. Special Logarithms
A logarithm with base 10 is called a common logarithm.
If no base is written for a logarithm, the base is assumed to be 10.
For example, log 5 = log105.

15. Special Logarithms
A logarithm with a base of e is called a natural logarithm and is abbreviated as “ln” (rather than as loge). Natural logarithms have the same properties as log base 10 and logarithms with other bases.
The natural logarithmic function f(x) = ln x is the inverse of the natural exponential function f(x) = ex.

16. Special Logarithms Common Logarithm log 10 𝑥 = log 𝑥
Natural logarithm (ln)
log 𝒆 𝒙 = ln 𝒙
Most calculators have keys for evaluating common and
natural logarithms.

17. 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

18. Your Turn:
for Example 3
Evaluate the logarithm. Use a calculator. 7.
log 12
about 1.079
ANSWER 8.
ln 0.75
about –0.288
ANSWER

19. 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 =

20. 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

21. Your Turn:
for Example 4
WHAT IF? Use the function in Example 4 to estimate the wind speed near a tornado’s center if its path is 150 miles long. 9.
The wind speed near the tornado’s center is about 267 miles per hour.
ANSWER

22. Inverse Functions
A logarithmic function/operation is the inverse function/operation of an exponential function/operation.
That is, the logarithm function 𝑔 𝑥 = log 𝑏 𝑥 is the inverse of the exponential function 𝑓 𝑥 = 𝑏 𝑥 . This means that:
𝑔 𝑓 𝑥 = log 𝑏 𝑏 𝑥 =𝑥 𝑎𝑛𝑑 𝑓 𝑔 𝑥 = 𝑏 log 𝑏 𝑥 =𝑥
The logarithmic operation undoes the exponential operation, with the same bases, and the exponential operation undoes the logarithm operation, with the same bases.

23. ( ) 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

24. Your Turn:
for Example 5
Simplify the expression.
10. 8
log x
ANSWER x
11. 7
log
7–3x
–3x
ANSWER

25. Your Turn:
for Example 5
Simplify the expression.
12. 2
log
64x 6x
ANSWER
13.
eln20 20
ANSWER

26. 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.

27. Your Turn:
for Example 6
Find the inverse of
14.
y = 4 x 4
log
y = x
ANSWER
y = ln (x – 5).
Find the inverse of
15.
ANSWER
The inverse of y = ln (x – 5) is y =
ex + 5.

28. Graphing Logarithm Functions
Because logarithms are the inverses of exponents, the inverse of an exponential function, such as y = 2x, is a logarithmic function, such as y = log2x.
The domain and range of each function are switched.
The domain of y = 2x is all real numbers (R), and the range is {y|y > 0}. The domain of y = log2x is {x|x > 0}, and the range is all real numbers (R).
Also the asymptotes are switched.
For y = 2x the asymptote is the x-axis (y = 0) and for y = log2x the
asymptote is they-axis (x = 0).

29. Graphing Logarithm Functions

30. Graphing Logarithm Functions
A logarithm is an exponent, so the rules for exponents also apply to logarithms. You may have noticed the following properties in the earlier examples.

31. EXAMPLE 7
Graph logarithmic functions
Graph the function. a. y = 3
log x
SOLUTION
Plot two convenient points (x = 1 & x = b), such as (1, 0), (3, 1). 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.

32. EXAMPLE 7
Graph logarithmic functions
Graph the function. b. y =
1/2
log x
SOLUTION
Plot two convenient points (x = 1 & x = b), such as (1, 0), (1/2,1). 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.

33. Your Turn:
for Example 7
Graph the function. State the domain and range.
16. y = 5
log x
ANSWER
The domain is x > 0, and the range is all real numbers.

34. 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) and (2, 1). 2
log
STEP 2
Translate the parent graph left 3 units and up 1 unit. The translated graph passes through (–2, 1) and (–1, 2). The graph’s asymptote is x = –3. The domain is x > –3, and the range is all real numbers.

35. Your Turn:
for Example 8
Graph the function. State the domain and range.
17. y =
1/3
log
(x – 3)
domain: x > 3,
range: all real numbers
ANSWER

36. Your Turn:
for Example 8
Graph the function. State the domain and range.
18. y = 4
log
(x + 1) – 2
domain: x > –1,
range: all real numbers
ANSWER