Name
January 4, 2017
Math 4 notes and problems
page 1
Arithmetic properties of logarithms
Objective: Prove and use three arithmetic properties of logarithms.
For each of the three major exponent properties, there’s a corresponding fact about logarithms.
Exponent property
bm · bn = bm + n
Logarithm property (assuming b > 0, b ≠ 1)
logb (RS) = logb R + logb S
“The log of a product equals a sum of logs.”
m
b
= b m−n
n
b
(bm)n = bmn
logb ( ) = logb R – logb S
R
S
“The log of a fraction equals a difference of logs.”
logb (Rc) = c · logb R
“The log of a power can be turned into a multiplication.
Just move the exponent c down in front of the log.”
Examples: using the properties
A. You can use the properties to break logs into pieces joined by +, –, and/or · :
log10 6 = log10 (2·3) = log10 2 + log10 3
log3 ( 7x ) = log3 7 – log3 x
ln (59) = 9 · ln 5
[notation reminder: ln( ) means loge( ) where e ≈ 2.718]
B. You can use the properties in the other direction to combine logs into a single log:
log10 2 + log10 3 = log10 (2·3) = log10 6
log3 7 – log3 x = log3 ( 7x )
log2 5 + log2 3 – log2 7 = log2 15 – log2 7 = log2 ( 157 )
For more examples of using the properties, see section 3.4 page 284 in the textbook.
Example: where the “log of a product” property comes from
Consider this multiplication using an exponent property:
23 · 24 = 27
8 · 16 = 128
Each of the powers involved can be rewritten as a log statement:
23 = 8 ⇒ log2 8 = 3
24 = 16 ⇒ log2 16 = 4
27 = 128 ⇒ log2 128 = 7
Now of course we know that 3 + 4 = 7. Putting that in terms of the logs:
3 + 4
= 7
log2 8 + log2 16 = log2 128
log2 8 + log2 16 = log2 (8 · 16)
This was just a single example, but if you tried the same thing with any other set of numbers,
it would work out the same way: two logs added together, equaling the log of a product.
Name
January 4, 2017
Math 4 notes and problems
page 2
Class Problems:
1. Break each of these logs into pieces joined by +, –, and/or ·.
(If you need some examples to follow, see part A of the examples on page 1.)
a. log5 ( 34 )
b. log6 (5 · 7)
c. log10 (28)
d. log2 (35)
Hint: First rewrite 35 as a multiplication.
e. log4 (3n)
f.
⎛x⎞
log 3 ⎜⎜ ⎟⎟
⎝ y⎠
g. log2 (3 · 45)
h. log10
⎛ cn
⎜⎜
⎝d
⎞
⎟⎟
⎠
Hint: Two steps can be taken here.
Name
January 4, 2017
Math 4 notes and problems
page 3
2. For each of the following, combine into a single log.
(If you need some examples to follow, see part B of the examples on page 1.)
a. log4 10 + log4 3
b. log7 30 – log7 5
c. 8 log10 2
Hint: This is like problem 1c but in reverse.
d. 3 log2 5
e. log3 2 + log3 4 + log3 5 – log3 7
f. log3 v – log3 w
g. log2 7 + log2 x
h. 3 log10 4 + log10 2
i. j log6 k – r log6 s
Hint: Two steps.
Name
January 4, 2017
Math 4 notes and problems
page 4
3. Combine into a single log, then figure out the value without a calculator.
Example: log5 50 – log5 2 = log5 ( 502 ) = log5 25 = 2
a. log6 12 + log6 3
b. log2 20 – log2 5
c. 6 log4 2
d. log10 2 + log10 25 + log10 20
e. log3 2 – log3 18
f. log12 6 + log12 10 – log12 5
g. log4 2 – log4 8
h. log6 9 + 2 log6 2
Homework Assignment
Do section 3.4 exercises 1-21 odd, also 10, 12, 20, 22, 59, and 61.
Name
January 4, 2017
Math 4 notes and problems
page 5
Answers to class problems:
Problem 1:
Problem 3:
a. log 5 3− log 5 4
a. 2
b. log 6 5 + log 6 7
b. 2
c. 8log10 2
d. log 2 5 + log 2 7
e. log 4 3+ log 4 n
c. 3
d. 3
e. −2
f. 1
f. log3 x − log3 y
g. −1
g. log 2 3+ 5log 2 4
h. 2
h. n log c − log d
See back of book for answers to odd
book problems.
Problem 2:
a. log 4 30
b. log 7 6
c. log 256
d. log 2 125
⎛ 40 ⎞
e. log3 ⎜ ⎟
⎝ 7⎠
⎛v⎞
f. log3 ⎜ ⎟
⎝w⎠
g. log 2 ( 7x )
h. log ( 2 ⋅ 43 )
⎛kj ⎞
i. log 6 ⎜ r ⎟
⎝s ⎠