Pre-AP Algebra 2
Unit 9 - Lesson 3 – Properties of Logarithms
Objectives:
 Students will be able to expand and condense logarithmic expressions using the properties of logs.
 Students will be able to prove the properties of logs.
Materials: Hw #9-2 answers overhead; pair work and answers overhead; homework #9-3
Time
20 min
Activity
HW Review
Put up answers to lesson #9-2 on the overhead; students check.
10 min
Do Now
Students work on check for understanding problems over homework.
Direct Instruction:
30 min
Background:
If powers with same base are equal, then exponents are equal.
317  32x7
1)
2)
25  81 x
If Logs with the same base are equal, then their arguments are equal.
3) log 2 16  log 2 2x
Concepts:
Special Cases (Matching Bases):
logb M
log b b M and b
log b b M  x
blogb M  x
rewrite
rewrite
b b
log b x  log b M
x
M
xM
xM
 log b b  M
M
 b logb M  M
Examples:
Simplify special cases:
4)
5)
6)
7)
6
log6 17
log5 52x
4
log 4 7
log10kt
Explore More Properties:
(Match the following)
1. log 4  log 2
2. log 4  log 5
3. log 6  log 2
4. 2 log 3
5. 3log 2
A. log 6
B. log 9
C. log 4
D. log 8
E. l og 3
F. log 20
Note how background info is used…
Concepts:
The 3 Log Properties
1) log b A  log b C 
2) log b A  log b C 
3) C log b A 
Condense:
8) log(6)  2log(2)  log(3)
9)
10)
log4 20  4log 4 x  log 4 y
log 2 20  2log 2 x  log 2 3  3log 2 y
Expand:
11) log(3x 4 )
12) log 5 2 x
y2
13) log 7
20 min
7 x
y2 z
Pair Work
Students work on the Properties of Logs worksheet.
Homework #9-3: Properties of Logarithms
Pre-AP Algebra 2
9-3 DO NOW
Name: __________________________
DO NOW
9-2 Check for Understanding
Evaluate without a calculator
1)
2)
4)
3)
Solve for
5)
7)
5 – Exemplary
4 – Proficient
3 – Nearly Proficient
2 – Emerging
1 – Beginning
(
in each equation
6)
)
Explain in your own words what is a logarithm?
8)
Pre-AP Algebra 2
9-3 Pair Work
Name: __________________________
Properties of Logarithms
1) Use logt A  0.25 and logt B  1.5 to find the exact value of each expression:
B
A
a. logt AB
b. log t
c. logt A3
d. logt A3 B
2) Simplify each expression.
a. log 8 8 9
b. log10 3x  7
c. 7log7 13
d.  log 1000
3) Expand each expression using the properties of logs.
a. log 2 9x
c. log 7
6x 2
y
b. log6 x5
d. log5
 
3
2
x
Hint: rewrite as x to a power…
4) Condense each expression using the properties of logs.
a. log5 8  log5 12
b. 2log x  log5


c.
1
log81 log3
2
1
1
d. 2 log6 15  log6 5  log6
2
25
e.
1
2
log81 log y
2
3
f. log5 (x  1)  log5 (x  3)
1
g. log x  3log y  log z
5


h. log 2x  5x 2  3log x
What went wrong?! For each answer, identify the line number that contains a mistake and
explain in words what the problem is. Then, write out the correct steps to the side.
1)
Correct Solution:
log16 2 + log16 4 + 5log16 2
line 1: = log16 8  log16 2 5
line 2: = log16 8  log16 32
line 3: = log16 40
line 4:
 1.204
Mistake in line # ____
Description of mistake:
2)
Correct Solution:
2log14 2 + 2log14 7
line 1: = 4 log14 14
line 2: = log14 14 4
line 3: = 4
Mistake in line # ____
Description of mistake:
3)
log y 2 + log y 3 = 36
line 1: log y 6  36
line 2: 6 y  36
line 3: y  2
Mistake in line # ____
Description of mistake:
Correct Solution:
Pre-AP Algebra 2
Homework #9-3
Name: __________________________
HW #9-3: Properties of Logarithms
Part 1: Multiple Choice
 4
1) What is the first incorrect step in simplifying log 4   ?
 64 
 4
step1 : log 4    log 4 4  log 4 64
 64 
step 2 :
 1  16
step 3 :
 15
a. Step 1
b. Step 2
c. Step 3
d. Each step is correct
2) log640 =
a. log10 6  log10 40
b. log10 6  log10 40
log10 40
d.
log10 6
c. log10 6 log10 40 
3) log 5 + 2 log x – ½ log y =
 y
a. log  2 
 5x 
 5x 2 
b. log 

 1 y 
2
 5x 
c. log  
 y
 5x 2 
d. log 

 y
4) What is the solution to the equation: 12 = 5 (log3 x) + 2
a. x = 9
b. x = 11
c. x = 2
d. x = 27
Part 2: Logs, Logs, Logs
Find the exact value of each logarithm. Show all work on your own paper – don’t use a calculator.
1) log 7 70.5 x
2) log10 3/ 4
3) 6 log6 9
4) 10 log y
5) log 4 2  log 4 32
6) log8 16  log8 2
log 9log2 7 
7) 2 2
8) 10log 6log 5
Expand the following logarithms into a sum and difference of logs. Change all powers into
coefficients.
 1 
9) loga x 2 y 3/ 4
10) log 
 a 
 ab 4 
11) log 6  3 
12) log x 2 1  x
2
 c 




 a
14) log 2  2 
b 
 
13) log a u 2v 3
Condense each expression into a single logarithm by using the properties of logs.
15) log3 16  log3 4
16) 2log x  log5
17) 7 log4 2  5log4 x  3log4 y
1
18) log3 2  log3 y
2
19)


1
1
log5 81  2log5 6  log5 4
4
2


 x 
 x  1
 log 
20) log 

 x  1
 x 
Solve each exponential equation. Find the exact answer and then use the change of base theorem
and your calculator to estimate the answer to the thousandths place. Plug in your estimate to the
original equation to check your result.
   40  650
21) 1000(2x )  450  350
22) 2 x
3
Part 3: STAAR Practice
1) A parabola is translated from f(x) = 4(x – 2)2 – 6 to f(x) = 4(x + 5)2 – 1. Describe the change.
a.7 right and 5 down
b. 7 left and 5 up
c. 3 left and 5 up
d. 3 right and 5 down
2) The formula
(√ ) can be used to approximate the period of a pendulum, where
is the
pendulum’s length in feet and is the pendulum’s period in seconds. If a pendulum’s period is
1.6 seconds, which of the following is closest to the length of the pendulum?
b. 1.4 ft
b. 4.2 ft
c. 2.1 ft
d. 3.2 ft
3) A chemical compound’s concentration in milligrams per liter during a reaction can be modeled by
the function below, where represents the number of seconds that have elapsed during the
reaction.
( )
In this situation, what are the domain and range for this function?
( )
a. Domain:
; range:
c. Domain:
b.Domain:
; range: ( )
d.Domain:
( )
; range:
; range: ( )
Lesson Name: Properties of Logarithms
Date: _______________ Student: ________________________
Concepts
Examples
Concepts:
Special Cases (Matching Bases):
log b b M and
Examples:
1) Simplify special cases:
logb M
b
log b b M  x
blogb M  x
rewrite
rewrite
b x  bM
log b x  log b M
xM
xM
 log b b  M
M
b
logb M
M
Note how background info is used…
2. log 4  log 5
3. log 6  log 2
4. 2 log 3
5. 3log 2
log6 17
1)
6
2)
log5 52x
3)
4)
4
1) 317  32x 7
2) 25  81 x
log 4 7
log10
kt
If Logs with the same base are equal, then
their arguments are equal.
3) log2 16  log2 2x  4
Explore More Properties:
(Match the following)
1. log 4  log 2
Background
If powers with same base are equal, then
exponents are equal.
A. log 6
B. log 9
C. log 4
D. log 8
E. l og 3
F. log 20
Concepts
Concepts:
Examples
Condense:
2) The 3 Log Properties
1) log b A  log b C 
2) log b A  log b C 
1.
log(6)  2log(2)  log(3)
2.
log4 20  4log 4 x  log 4 y
3.
log 2 20  2log 2 x  log 2 3  3log 2 y
3) C log b A 
proof of multiplication property
Let logb A  M and logb C  N
Convert : b M  A and b N  C
Expand:
4.
log(3x 4 )
5.
log 5
2x
y2
6.
log 7
7 x
y2 z
log b AC  log b (b M b N )
 log b (b M  N )
=M N
 logb A  logb C
 logb AC  logb A  logb C