MA 15800
Lesson 16 Notes
Properties of Logarithms
Summer 2016
Remember: A logarithm is an exponent! It ‘behaves’ like an exponent!
In the last lesson, we discussed four properties of logarithms.
1) log a 1  0
2) log a a  1
3) log a a x  x
4) a loga x  x
This lesson covers more properties for logarithms. Since logarithms ‘behave’
like exponents, you should see a similarity between the properties of
exponents and the properties of logarithms.
I
PRODUCT RULE
Product Rule for Exponents: bm bn  bmn
(when powers are multiplied, exponents are added)
Product Rule for Logarithms: logb MN  logb M  logb N
(the logarithm of a product is the sum of the logarithms)
Informal proof of product rule:
𝑙𝑜𝑔100 = 2 𝑙𝑜𝑔1000 = 3
log(100 1000)
 log100000
5
 23
 log100  log1000
When a single logarithm is written using the product rule, we refer to this process as expanding the
logarithm or expanding the logarithmic expression. (Because we can only find logarithms of
positive values; when using these properties of logarithms, we assume all variables represent positive
values.
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MA 15800
Lesson 16 Notes
Properties of Logarithms
Summer 2016
Ex 1: Use the product rule to expand each expression and simplify where possible.
a ) log 2  7 r  
b) log b (2 x 2 y ) 
log(100ab) 
d ) ln(20e5 ) 
II
QUOTIENT RULE
bm
 bmn
n
b
(when powers are divided, exponents are subtracted)
M 
Quotient Rule for Logarithms: logb    logb M  logb N
N
(the logarithm of a quotient is the difference of the logarithms of the numerator and the denominator)
Quotient Rule for Exponents:
CAUTION:
log b ( M  N )  log b M  log b N
log b
M log b M

N log b N
We can also expand a logarithm using the quotient rule. Again, all variable are assumed to represent
positive values.
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MA 15800
Lesson 16 Notes
Properties of Logarithms
Summer 2016
Ex 2: Use the quotient rule to expand each logarithm and simplify where possible.
9
a ) log 3   
Note: I used parentheses
 y
around the argument on
the logarithms at the left.
Some textbooks may not
include the parentheses.
3
log b   is equivalent
 x 
b) log 
m

 1000 
3
to log b
m
III
Power Rule
Power Rule for Exponents:
b 
m n
 bmn
(when a power is raised to another power, the exponents are multiplied)
Likewise, when a logarithm’s argument has an exponent, the exponent is multiplied by the logarithm.
Power Rule for Logarithms: logb M p  p logb M
The logarithm of a power is the product of the exponent and the logarithm.
We can also use the power rule to expand a logarithm. (Assume all variables represent positive
values.)
Ex 3: Use the power rule to expand each logarithm and simplify where possible.
a ) log x8 
b) log 5 (253 ) 
c) ln y 
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MA 15800
IV
Lesson 16 Notes
Properties of Logarithms
Summer 2016
Summary of properties of LOGARITHMS
All variables represent positive values and all bases are positive numbers
other than 1.
1) log b 1  0
logarithm of 1 is zero
2) log b b  1
3) log b b x  x
4) b logb x  x
5) log b ( MN )  log b M  log b N
M
6) log b 
N

  log b M  log b N

7) log b  M p   p log b M
Product Rule
Quotient Rule
Power Rule
Ex 4: Use the properties of logarithms to expand each logarithmic expression. Assume all variables
represent positive values. (Express each in terms of logarithms of x, y, and/or z.)
 xy 
a ) log 

 z
 4 x3 
b) ln  5  
 yz 
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MA 15800
Lesson 16 Notes
Properties of Logarithms
Summer 2016
c) log 2 3 xy 


d ) log 3 27 x 2 3 y 
The reverse of expanding a logarithm or a logarithmic expression is condensing a logarithmic
expression or writing the expression as one logarithm. Again, assume all variables represent positive
values.
Hint: It is helpful to arrange the subtraction at the end of the line and then factor out the negative or
subtraction sign!!
Ex 5: Condense each logarithm or logarithmic expression. Write as a single logarithm.
1
2
a ) log 3  log x  2 log y  log z 
b)
1
log( x  2)  2 log x  2 log 4 
3
c)
1
(ln x  3ln
2
y )  3ln( x  2)
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MA 15800
Lesson 16 Notes
Properties of Logarithms
Summer 2016
x
d ) log( x 2 y 2 )  3log x y  2log   
 y
Hint: For the example above, it might be wise to expand where possible first, and then condense into
one logarithm.
Ex 6: Use the properties of logarithms and the information below to evaluate each logarithm.
logb m  2.3892, logb n  1.2389, and logb r  0.8881
a ) log b (m 2 n) 
 n
b) log b 
 
 r 
Ex 7: Use the properties of logarithms and the logarithms below to find the following values.
logb 8  1.8928, logb 11  2.1827, and logb 2  0.6309
a) logb 4 
(problem 7 continued on the next page)
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MA 15800
Lesson 16 Notes
Properties of Logarithms
Summer 2016
b) log b 88 
c) log b 121 
d ) log b 44 
Ex 8: Let log 2 4  A and log 2 5  B . Write each expression in terms of A and/or B.
a ) log 2 125 
b) log 2
5

4
Ex 9: Use the properties of logarithms to assist in solving these equations.
a) log 4 (3 x  2)  log 4 5  log 4 3
1
2
b) ln x  ln( x  6)  ln 9
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MA 15800
c)
Lesson 16 Notes
Properties of Logarithms
Summer 2016
log 3 ( x  3)  log 3 ( x  5)  1
d ) log 3 ( x  2)  log 3 27  log 3 ( x  4)  5log5 1
Ex 10: When the volume control on a stereo system is increased, the voltage across a loudspeaker
V
changes from V1 to V2, and the decibel increase in gain is given by db  20 log 2 . Approximate the
V1
decibel increase if the voltage changes from 2 volts to 4.5 volts.
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