Properties of Logarithms
Section 3.3
Objectives
• Rewrite logarithms with different bases.
• Use properties of logarithms to evaluate or
rewrite logarithmic expressions.
• Use properties of logarithms to expand or
condense logarithmic expressions.
Logarithmic FAQs
• Logarithms are a mathematical tool originally invented to
reduce arithmetic computations.
• Multiplication and division are reduced to simple addition
and subtraction.
• Exponentiation and root operations are reduced to more
simple exponent multiplication or division.
• Changing the base of numbers is simplified.
• Scientific and graphing calculators provide logarithm
functions for base 10 (common) and base e (natural) logs.
Both log types can be used for ordinary calculations.
Logarithmic Notation
• For logarithmic functions we use the
notation:
loga(x) or logax
• This is read “log, base a, of x.” Thus,
y = logax means x = ay
• And so a logarithm is simply an exponent
of some base.
Remember that to multiply powers with the
same base, you add exponents.
Adding Logarithms
Express log64 + log69 as a single
logarithm.
Are the bases the same?
Simplify.
To add the logarithms, multiply log6 (4  9)
the numbers.
log6 36
Simplify.
Think: 6? = 36.
Or convert to a base of 6 and
solve for the exponent.
log64 + log69 =
6𝑦 = 62
2
Express as a single logarithm.
Simplify, if possible.
log5625 + log525
Are the bases the same?
To add the logarithms, multiply
the numbers.
log5 (625 • 25)
Simplify.
log5 15,625
Think: 5? = 15625
Convert to a base of 5 and
solve for the exponent.
log5625 + log525 = 6
5 𝑦 = 56
Express as a single logarithm.
Simplify, if possible.
log 1 27 + log 1
3
3
1
9
Are the bases the same?
To add the logarithms, multiply log 1 (27 •
3
the numbers.
1
9
)
log 1 3
Simplify.
3
Think:
1 ?
3 =
3
1
Convert to a base of
and
3
solve for the exponent.
1
log 1 27 + log 1 = –1
3
39
1 𝑦
( ) = 31
3
1 𝑦
( )
3
=
1 −1
( )
3
Remember that to divide powers with the
same base, you subtract exponents
Because logarithms are exponents, subtracting
logarithms with the same base is the same as
finding the logarithms of the quotient with that
base.
The property above can also be used in reverse.
Caution
Just as a5b3 cannot be simplified, logarithms
must have the same base to be simplified.
Express log5100 – log54 as a single
logarithm.
Simplify, if possible.
log5100 – log54
Are the bases the same?
To subtract the logarithms,
divide the numbers.
log525
Simplify.
Think: 5? = 25.
log5100 – log54 =
log5(100 ÷ 4)
2
Express log749 – log77 as a single
logarithm.
Simplify, if possible.
log749 – log77
Are the bases the same?
To subtract the logarithms,
divide the numbers
log7(49 ÷ 7)
Simplify.
log77
Think: 7? = 7.
log 7 49 − log 7 7 = 1
Because you can multiply logarithms, you can
also take powers of logarithms.
Express as a product.
Simplify, if possible.
A. log2326
B. log8420
6log232
20log84
5
8𝑦 = 4
Because 2 = 32,
log232 = 5.
23
6(5) = 30
23𝑦 = 22
6log232 = 30
2
𝑦=
3
𝑦
= 22
20(
2
3
)=
log 8
420
=
40
3
𝟒𝟎
𝟑
Express as a product.
Simplify, if possibly.
log104
log5252
4log10
2log525
Because
101 = 10,
log 10 = 1.
Because
52 = 25,
log525 = 2.
4(1) = 4
2(2) = 4
log10
4
=𝟒
log525
2
=𝟒
Express as a product.
Simplify, if possibly.
log2 (
1
2
5log2 (
1
2𝑦
2
)5
)
Solve
1
=
2
1
log 2
2
2𝑦 = 2−1
𝑦 = −1
log 2
1
2
5(–1) = –5
5
=
−𝟓
The Product Rule of Logarithms
Product Rule of Logarithms
If M, N, and a are positive real numbers, with
a  1, then loga(MN) = logaM + logaN.
Example: Write the following logarithm as a sum of
logarithms.
(a) log5(4 · 7)
log5(4 · 7) = log54 + log57
(b) log10(100 · 1000)
log10(100 · 1000) = log10100 + log101000
=2+3=5
Your Turn:
• Express as a sum of
logarithms:
2
log3 ( x w)
Solution:
log3 ( x w)  log3 x  log3 w
2
2
The Quotient Rule of Logarithms
Quotient Rule of Logarithms
If M, N, and a are positive real numbers, with
a  1, then log  M   log M  log N.
a

N 
a
a
Example: Write the following logarithm as a
difference of logarithms.
10
(a) log5   = log5 10  log5 3
 3
 c
(b) log8    log8 c  log8 4
 4
Your Turn:
• Express as a difference
of logarithms.
10
log a
b
• Solution:
10
log a  log a 10  log a b
b
Sum and Difference of Logarithms
 8y
log
Example: Write
as the sum or difference
6
 5 
of logarithms.
 8y  log (8 y)  log 5
log6  
Quotient Rule
6
6
 5
 log 6 8  log 6 y  log 6 5 Product Rule
The Power Rule of Logarithms
The Power Rule of Logarithms
If M and a are positive real numbers, with a
 1, and r is any real number,
then loga M r = r loga M.
Example: Use the Power Rule to express all
powers as factors.
log4(a3b5) = log4(a3) + log4(b5)
= 3 log4a + 5 log4b
Product Rule
Power Rule
Your Turn:
• Express as a product.
log a 7
3
Solution:
3
log a 7  3log a 7
Your Turn:
• Express as a product.
5
loga 11
• Solution:
log a 11  log a 11
5
1/5
1
 log a 11
5
Rewriting Logarithmic Expressions
• The properties of logarithms are useful for rewriting
logarithmic expressions in forms that simplify the
operations of algebra.
• This is because the properties convert more
complicated products, quotients, and exponential
forms into simpler sums, differences, and products.
• This is called expanding a logarithmic expression.
• The procedure above can be reversed to produce a
single logarithmic expression.
• This is called condensing a logarithmic
expression.
Examples:
• Expand:
• log 5mn =
• log 5 + log m + log n
• Expand:
• log58x3 =
• log58 + 3·log5x
Expand – Express as a Sum and
Difference of Logarithms
7x
• log2 =
y
3
• log27x3 - log2y =
• log27 + log2x3 – log2y =
• log27 + 3·log2x – log2y
Condensing Logarithms
• log 6 + 2 log2 – log 3 =
• log 6 + log 22 – log 3 =
• log (6·22) – log 3 =
• log 6  2 =
2
3
• log 8
Examples:
• Condense:
• log57 + 3·log5t =
• log57t3
• Condense:
• 3log2x – (log24 + log2y)=
3
x
• log2
4y
Your Turn:
• Express in terms of sums and
differences of logarithms.
3
w y
log a 2
z
4
• Solution:
3
4
w y
3 4
2
log a 2  log a ( w y )  log a z
z
 log a w3  log a y 4  log a z 2
 3log a w  4log a y  2log a z
Change-of-Base Formula
Only logarithms with base 10 or base e can be found by using a
calculator. Other bases require the use of the Change-of-Base
Formula.
Change-of-Base Formula
If a  1, and b  1, and M are positive real numbers, then
log b M
log a M 
.
log b a
Example:
Approximate log4 25.
log 4 25 
10 is used for
both bases.
log10 25 log 25 1.39794  2.32193


log 4 0.60206
log10 4
Change-of-Base Formula
Example:
Approximate the following logarithms.
(a) log3 198
log198 2.297
log 3 198 

 4.816
log 3
0.477
(b) log6 5
log 6
log 5  0.349  0.449
5
0.778
log 6
Your Turn:
Evaluate each expression and round to four
decimal places.
(a) log 5 17
Solution
(a) 1.7604
(b) -3.3219
(b) log 2 .1