Logarithm Rules: Product, Quotient, and Power Rule with Examples, Slides of Algebra

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§9.4a

Logarithm Rules

Review §

 Any QUESTIONS About

• §9.3 → Common & Natural Logs

 Any QUESTIONS About HomeWork

• §9.3 → HW-

9.3 MTH 55

Quotient Rule for Logarithms

• Let M , N , and a be positive real numbers with a ≠ 1, and let r be any real number. Then the QUOTIENT Rule

 That is, The logarithm of the quotient of two (or more) numbers is the difference of the logarithms of the numbers

log a

M

N











 =^ log a M^ −^ log a N

Power Rule for Logarithms

• Let M , N , and a be positive real numbers with a ≠ 1, and let r be any real number. Then the POWER Rule

 That is, The logarithm of a number to the power r is r times the logarithm of the number.

log a M

r = r log a M

Example  Product Rule

• Express as an equivalent expression that is a single logarithm: log a 6 + log a 7
• Solution

= log a (42). Using the productrule for logarithms

log a 6 + log a 7 = log a (6·7)

Example  Quotient Rule

• Express as an equivalent expression that is a single logarithm: log 3 (9/ y )
• Solution

log 3 (9 /y ) = log 3 9 – log 3 y. Using the quotientrule for logarithms

Example  Power Rule

• Use the power rule to write an equivalent expression that is a product: a) log a 6 −^3 b) log 4 x.

 Solution

= log 4 x 1/

Using the power a) log a (^6) rule for logarithms − (^3) = −3log a 6

b) log 4 x

= ½ log 4 x Using the powerrule for logarithms

Example  Use The Rules

• Given that log 5 z = 3 and log 5 y = 2, evaluate each expression. a. log (^5) ( yz ) b. log (^5) ( 125 y^7 )

c. log (^5)

z y

d. log 5 z

1  (^30) y 5 

a. log 5 ( yz ) = log 5 y + log 5 z = 2 + 3 = 5

 Solution

Example  Use The Rules

• Soln

d. log 5 z

1

 30 y 5

= log 5 z

1

30 + log 5 y^5

log 5 z + 5 log 5 y

( )^3 +^ 5 2( )

Example  Use The Rules

• Express as an equivalent expression using individual logarithms of x , y , & z

 Soln a)

3 a) log 4 x^ b) log b^3 xy 7 yz (^) z

= log 4 x^3 – log 4 yz = 3log 4 x – log 4 yz = 3log 4 x – (log 4 y + log 4 z )

= 3log 4 x –log 4 y – log 4 z

3 log (^4) x yz

Caveat on Log Rules

• Because the product and quotient rules

replace one term with two, it is often

best to use the rules within parentheses,

as in the previous example

( )

1 log log 7log 3

= (^) b x + (^) b y − b z

7

(^1) log 3 b

xy z

= ⋅

Example  Expand by Log Rules

• Write the expressions in expanded form

a. log (^2)

x^2 ( x − (^1) )^3 (^2 x^ +^1 )^4

b. log c x^3 y^2 z^5

 Solution a)

log (^2)

x^2 ( x − (^1) )^3 (^2 x^ +^1 )^4

= log 2 x^2 ( x − (^1) )^3 − log (^2) ( 2 x + (^1) )^4

= log 2 x^2 + log (^2) ( x − (^1) )^3 − log (^2) ( 2 x + (^1) )^4 = 2 log 2 x + 3log (^2) ( x − (^1) ) − 4 log (^2) ( 2 x + (^1) )

Example  Condense Logs

• Write the expressions in condensed form

a. log 3 x − log 4 y

b. 2 ln x +

ln (^) ( x 2 + (^1) )

c. 2 log 2 5 + log 2 9 − log 2 75

d.

 ln x + ln (^) ( x + (^1) ) − ln (^) ( x 2 + (^1) )

Example  Condense Logs

• Solution a) log 3 x − log 4 y = log

3 x 4 y

 

 

 Solution b)

2 log x +

1 2

ln (^) ( x 2 + (^1) )= ln x^2 + ln (^) ( x 2 + (^1) )

1 2

= ln (^) ( x 2 x^2 + (^1) )