# 9.5 Properties of Logarithms

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9.5 Properties of 9.5 Properties of LogarithmsLogarithms

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Laws of LogarithmsLaws of Logarithms

Just like the rules for exponents there are corresponding rules for logs that allow you to rewrite the log of a product, the log of a quotient, or the log of a power.

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Log of a ProductLog of a Product

Logs are just exponents The log of a product is the sum of the logs of

the factors: logb xy = logb x + logb y

Ex: log (25 ·125) = log 25 + log 125

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Log of a QuotientLog of a Quotient

Logs are exponents The log of a quotient is the

difference of the logs of the factors: logb = logb x – logb y

Ex: ln ( ) = ln 125 – ln 25

x

y125

25

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Log of a PowerLog of a Power

Logs are exponents The log of a power is the product of

the exponent and the log: logb xn = n∙logb x Ex: log 32 = 2 ∙ log 3

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Rules for LogarithmsRules for Logarithms

These same laws can be used to turn an expression into a single log:

logb x + logb y = logb xy

logb x – logb y = logb

n∙logb x = logb xn

x

y

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logb(xy) = logb x + logb y

Express 3logAB

C

as a sum and difference of logarithms:

3logAB

C

= log3A + log3B

ExamplesExamples

logb( ) = logb x – logb y logb xn = n logb x_______________________________

Solve: x = log330 – log310

= log33

3

30= log

10

Evaluate: 5 25 125log1

25 525 125log log

5

1125

22 log

= 2 = 13

2

7

2

– log3C

x = 1

= log3 AB

x

y

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Sample ProblemSample Problem Express as a single logarithm:

3log7x + log7(x+1) - 2log7(x+2) 3log7x = log7x3

2log7(x+2) = log7(x+2)2

log7x3 + log7(x+1) - log7(x+2)2

log7(x3·(x+1)) - log7(x+2)2

log7(x3·(x+1)) - log7(x+2)2 =

log( )7

3

2

1

2

x x

x

b glog

( )7

3

2

1

2

x x

x

b g

To use a calculator to evaluate logarithms with other bases, you can change the base to 10 or “e” by using either of the following:

For all positive numbers a, b, and x, where a ≠ 1 and b ≠ 1:

Example: Evaluate log4 22

≈ 2.2295

b

a b

log xlog x

log a

Change of Base Formula

a

log xlog x

log a

a

ln xlog x

ln a22

224

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loglog

log1.3424

=0.6021