Math-1050Session #25

6.6: Properties of Logs(1) Product of Logs(2) Log of a Power(3) Log of a Quotient

Find )(1 xf −

425 += xy

42log 5 += yx

yx 2log4 5 =+−

yx=

+−

2

log4 5

425)( += xxf

425 += yx

xy 5log2

12+−=

Replace f(x) with ‘y’

exchange ‘x’ and ‘y’

Log is the exponent (remember how to

convert between the two?)

Solve for ‘y’

The product of powers → add the exponents

Logarithm: another way of writing the exponent

log2 8 + log2 4

Convert each exponent above into a log:

3 + 2 = 5

= log2 32

23 ∗ 22 = 25

This is the logarithm equivalent of the multiply powers property

of exponents.

23 ∗ 22 = 25

SRRS bbb loglog)(log +=

)5*3(log15log 22 =

Log of a Product Property

log of a product = sum of the logs of the factors.

5log3log15log 222 +=

xy3log yx 33 loglog +

45log 35log3log3log 333 ++

5*3*345 = 5log3log2 33 +

Expand the Logarithm: use properties of logs to rewrite a

single log as an expression of separate logs.

Expand the Logarithm: use properties of logs to rewrite a

single log as an expression of separate logs.

)3log( 2xy 2loglog3log yx ++=

yyx logloglog3log +++=

yx log2log3log ++=

6log 4

xyw2ln

2log3log 44 +=

wyx lnlnln2ln +++=

)5*7(log 2=

xlog5log +

5log7log 22 + 35log 2=

Condense the Logarithm: apply properties of logarithms to rewrite

the log expression as a single log.

x5log=

7log5log 57 + “unlike logs” → can’t condense

“Condense the Log”

7log2log 55 +

4log9log +

4log6log 85 +

14log 5=

36log=

“unlike logs” → can’t condense

9log2

3log3log 22 +=

“Expand the Log”

)3*3(log2=

3log2 2=

Notice something interesting

9log22

2 )3(log= 3log2 2=

Log of a Power Property

16log3

4log4log 33 +=

“Expand the Product”

)4*4(log3=

4log2 3=

Notice something interesting

16log3

2

3 )4(log= 4log2 3=

Log of a Power Property

10log2 5=

2

5 10log

2

5 10log

10log10log 55 +=

10log2 5=

“Expand the Product”

Log of a product is the sum of the logs

of the factors.

Combine “like terms”

New property: “log of a power”

3log x

Use Log of a Power simplify

8ln

xlog3=

32ln= 2ln3=

xlog 21

log x= xlog2

1=

Gotcha’

Which one?y3log5=

53log y5log3log y+=

y3log5 ( )53log y= 553log y=

Log of a Power

c

b Rlogc Rc blog→

A potential error is this:

3

2 6log x x6log3 2=

What is the error ? ‘3’ is an exponent of the base ‘x’ not ‘6x’

Correct the error.

3

22

3

2 log6log6log xx +=

x222 log32log3log ++=

3

2

y

x Using properties of exponents: rewrite this so

the ‘y’ term is NOT in the denominator.32 −yx

2

5log 3

2log)1(5log 33 −+=

( )1

3 2*5log −=

1

33 )2(log)5(log −+=

Negative Exponent Property

Log of a Product Property

Log of a Power Property

2log5log 33 −= Definition of Subtraction: (adding

a negative is subtraction)

Log of a Quotient Property

SRS

Rbbb logloglog −=

2

5log 3

2log5log 33 −“expand the quotient”

3ln8ln − “condense the quotient”3

8ln

“Negative Log”→ denominator of the logarand

16log8log 55 −

5

4log

Condense the quotient

2log5log 44 −

Expand the Quotient

7

3ln

5log4log −=

7ln3ln −=

5log2log2log −+=

5log2log2 −=

2

5log 4=

16

8log 5=

2

1log 5=

Expand the Logarithm

The

denominator

is a product!

53

2log

y

x

)log53(log2log yx +−=

yx log53log2log −−=

Distributive property!

yx log53loglog2log −−+=

53log2log yx −=

Logs of factors in the numerator will be positive.

Logs of factors in the denominator will be negative.

Expand the quotient

3

2

5ln

y

zx 32 5lnln yzx −=

)ln5(lnlnln 32 yxz +−+=

)ln35(lnln2ln yxz +−+=

yxz ln35lnln2ln −−+=2

7

5

4log

=

x

w14

10

4logx

w=

14

4

10

4 loglog xw −=

xw 44 log14log10 −=

yz

x

4

2log4 yzx 4log2log 44 −=

zyx 44444 loglog4loglog2log −−−+=

zyx 44444 loglog4loglog2

12log −−−+=

5log 44log

5log

10

10=

c

aa

b

bc

log

loglog =

Change-of-Base Formula for Logarithms Change to log base 10 or base ‘e’

(your calculator can do these).

Convert to base 10.

6021.0

699.0= 161.1=

4ln

5ln=5log 4

386.1

609.1= 161.1=

=log10 6

log10 3𝑙𝑜𝑔 3 6 =

𝑙𝑜𝑔612

𝑙𝑜𝑔 312

=

12𝑙𝑜𝑔6

12𝑙𝑜𝑔 3

=𝑙𝑜𝑔6

𝑙𝑜𝑔 3= 𝑙𝑜𝑔6 − 𝑙𝑜𝑔3 NO!!!!!

log6

𝑙𝑜𝑔3≠ log

6

3

All non-base 10 logs are vertical stretches of the base 10 log.

xy 2log=

2log

log xy =

change of base formula.

301.0

log x= xlog*32.3=

12log

2log2log 2 ==

x=2log2

1=x

22 =x

Simplify

2log2

Using Change of base:

Simplify: 16log4

2

4 4log 4log2 42)1(2 =

“4 raised to what power equals 16?”

2log2

21

2 2log 2log2

12

“2 raised to what power equals the square root of 2?”

)1(2

12

1

Simplify: 27log5 3

3

3 3log5 3log)5*3( 315

4)16(log6 2 −

4)2(log6 4

2 −

4)2(log)6*4( 2 −

4)1)(24( −

20

Simplify:

3)125(log8 5 +

5)81(log2 9 −

4)3(log6 23 +−

27=

1−=

1=