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.

Express log64 + log69 as a single logarithm.

Simplify.

Adding Logarithms

2

To add the logarithms, multiply the numbers. log6 (4 9)

log6 36Simplify.

Think: 6?

= 36.

log64 + log

69 =

6 𝑦=62Or convert to a base of 6 and solve for the exponent.

Are the bases the same?

Express as a single logarithm.

Simplify, if possible.

6

To add the logarithms, multiply the numbers.

log5625 + log525

log5 (625 • 25)

log5 15,625Simplify.

Think: 5

?

= 15625

Convert to a base of 5 and solve for the exponent. 5𝑦=56

log5625 + log

525 =

Are the bases the same?

Express as a single logarithm.

Simplify, if possible.

–1

To add the logarithms, multiply the numbers.

Simplify.

Think:

?

= 31

3

1

3

log (27 • )1

9

1

3

log 3

log 27 + log1

3

1

3

1

9

Convert to a base of and solve for the exponent.

(13)𝑦=31

(

log 1327+ log 1

3

19=¿

Are the bases the same?

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.

Just as a5b

3 cannot be simplified, logarithms must have the same base to be

simplified.

Caution

Express log5100 – log54 as a single logarithm.

Simplify, if possible.

To subtract the logarithms, divide the numbers.

log5100 – log54

log5(100 ÷ 4)

2

log525Simplify.

Think: 5

?

= 25.

Are the bases the same?

log5100 – log54 =

Express log749 – log77 as a single logarithm.

Simplify, if possible.

To subtract the logarithms, divide the numbers

log749 – log77

log7(49 ÷ 7)

1

log77Simplify.

Think: 7

?

= 7.

Are the bases the same?

log 7 49 − log77=¿¿

Because you can multiply logarithms, you can also take powers of logarithms.

Express as a product.

Simplify, if possible.

A. log2326

B. log8420

6log232

6(5) = 30

20log84

20( ) = 40

3

2

3

Because 2

5

= 32, log232 = 5.

6log232 = 30

8 𝑦=4

(23 )𝑦=22

23 𝑦=22

𝑦=23

log 8 420=¿𝟒𝟎𝟑

Express as a product.

Simplify, if possibly.

log104

log5252

4log10

4(1) = 4

2log525

2(2) = 4

Because 52

= 25,

log525 = 2.

Because 10

1

= 10,

log 10 = 1.

log104 ¿𝟒 log5252¿𝟒

Express as a product.

Simplify, if possibly.

log2 ( )5

5(–1) = –5

5log2 ( ) 1

2

1

2

log 2( 12 )

5

=¿¿

2𝑦=12

2𝑦=2−1

𝑦=−1

−𝟓

Solve

The Product Rule of Logarithms

Product Rule of LogarithmsIf M, N, and a are positive real numbers, with a 1, then loga(MN) = logaM + logaN.

(a) log5(4 · 7)log5(4 · 7) = log54 + log57

log10(100 · 1000) = log10100 + log101000= 2 + 3 = 5

Example: Write the following logarithm as a sum of logarithms.

(b) log10(100 · 1000)

Your Turn:

• Express as a sum of logarithms:

Solution:

2 23 3 3log ( ) log logx w x w

23log ( )x w

The Quotient Rule of Logarithms

Quotient Rule of LogarithmsIf M, N, and a are positive real numbers, with a 1, then

Example: Write the following logarithm as a difference of logarithms.

loga

MN

loga M loga N.

(a) log5

103

(b) log8

c4

= log510 log5 3

8 8log log 4c

Your Turn:• Express as a

difference of logarithms.

• Solution:

10loga b

10log log 10 loga a a bb

Sum and Difference of Logarithms

Example: Write as the sum or difference of logarithms.

log6

8y5

log6

8y5

6 6log (8 ) log 5y Quotient Rule

6 6 6log 8 log log 5y Product Rule

The Power Rule of Logarithms

Example: Use the Power Rule to express all powers as factors.

log4(a3b5)

The Power Rule of LogarithmsIf M and a are positive real numbers, with a 1, and r is any real number, then loga M r = r loga M.

= log4(a3) + log4(b5) Product Rule

= 3 log4a + 5 log4b Power Rule

Your Turn:• Express as a product.

Solution:

3log 7a

3log 7 3log 7a a

Your Turn:• Express as a product.

• Solution:

5log 11a

1/55log 11 log 111 log 115

a a

a

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

• log2 =

• log27x3 - log2y = • log27 + log2x3 – log2y =• log27 + 3·log2x – log2y

yx37

Condensing Logarithms

• log 6 + 2 log2 – log 3 =• log 6 + log 22 – log 3 =• log (6·22) – log 3 =

• log =

• log 8 326 2

Examples:

• Condense:

• log57 + 3·log5t =• log57t3

• Condense:

• 3log2x – (log24 + log2y)=

• log2 yx4

3

Your Turn:• Express in terms of sums and

differences of logarithms.

• Solution:3 4

3 4 22

3 4 2

log log ( ) log

log log log3log 4log 2log

a a a

a a a

a a a

w yw y z

zw y z

w y z

3 4

2logaw y

z

Change-of-Base Formula

Example:Approximate log4 25.

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 FormulaIf a 1, and b 1, and M are positive real numbers, then

loglog .logb

ab

MM a

104

10

log 25log 25 log 4 log25log4 1.39794

0.60206 2.32193

10 is used for both bases.

Change-of-Base Formula

Example:Approximate the following logarithms.

3log198log 198 log3 2.297

0.477 4.816

3log 198(a)

6log 5(b)

6log 5log 5 log 6 0.349

0.778 0.449

Your Turn:

Evaluate each expression and round to four decimal places.

(a)

Solution

(a) 1.7604

(b) -3.3219

17log5 1.log(b) 2