Properties of LogarithmsSection 3.3

Properties of LogarithmsWhat logs can we find using our calculators?

◦Common logarithm◦Natural logarithm

Although these are the two most frequently used logarithms, you may need to evaluate other logs at times

For these instances, we have a change-of-base formula

Properties of LogarithmsChange-of-Base Formula

Let a, b, and x be positive real numbers such that a ≠ 1 and b ≠ 1. Then can be converted to a different base as follows:

x Loga

aa Log xLog x Log

b

b

Base b

Properties of LogarithmsChange-of-Base Formula

Let a, b, and x be positive real numbers such that a ≠ 1 and b ≠ 1. Then can be converted to a different base as follows:

x Loga

aa Log xLog x Log

10

10

Base 10

Properties of LogarithmsChange-of-Base Formula

Let a, b, and x be positive real numbers such that a ≠ 1 and b ≠ 1. Then can be converted to a different base as follows:

x Loga

aa ln ln x x Log

Base e

Properties of LogarithmsEvaluate the following logarithm:

30 Log4

→ 4 raised to what power equals 30?Since we don’t know the answer to this, we would want to use the change-of-base formula

4 Log30 Log 30 Log4 2.4534

Properties of LogarithmsEvaluate the following logarithm using

the natural log function.

14 Log2

14 Log2 0.693152.63906

2ln 14ln 3.8073

Properties of LogarithmsEvaluate the following logarithms using

the common log and the natural log.

a)

b)

18 Log5

42 Log2

.79591

3923.5

Properties of LogarithmsWhat is a logarithm?

Therefore, logarithms should have properties that are similar to those of exponents

Properties of LogarithmsFor example, evaluate the following:

a)

b)

c)

52 x x 5 2x 7x

37 x x 3 7x 4x

27 )(x 2 7x 14x

Properties of LogarithmsJust like we have properties for

exponents, we have properties for logarithms.

These properties are true for logs with base a, the common logs, and the natural logs

Properties of LogarithmsProperties of LogarithmsLet a be a positive number such that a ≠ 1, and

let n be a real number. If u and v are positive real numbers, the following properties are true.

(uv)Log 1) a vLog u Log aa

vuLog 2) a vLog u Log aa

nu Log 3) a u Logn a

Properties of LogarithmsUse the properties to rewrite the

following logarithm:

10zLog3

From property 1, we can rewrite this as the following:

10zLog3 10Log3 z Log3

Properties of LogarithmsUse the properties to rewrite the

following logarithm:

2y Log10

From property 2, we can rewrite this as the following:

2yLog10 y Log10 2 Log10

Properties of LogarithmsUse the properties to rewrite the

following logarithm:

36 z1 Log

From property 3, we can rewrite this as the following:

36 z1Log z Log -3

6 z Log3 6

Properties of LogarithmsSection 3.3

Properties of LogarithmsYesterday:

a) Change-of-Base Formula

b) 3 Properties

Properties of LogarithmsToday we are going to continue working

with the three properties covered yesterday.

(uv)Log 1) a vLog u Log aa

vuLog 2) a vLog u Log aa

nu Log 3) a u Logn a

Properties of LogarithmsThese properties can be used to rewrite

log expressions in simpler terms

We can take complicated products, quotients, and exponentials and convert them to sums, differences, and products

Properties of LogarithmsExpand the following log expression:

y5xlog 34

Start by applying property 1 to separate the product:

y5xlog 34 5log4 3

4xlog ylog4

Properties of LogarithmsExpand the following log expression:

Apply property 3 to eliminate the exponent

y5xlog 34 5log4 3

4xlog ylog4

5log4 xlog3 4 ylog4

Properties of LogarithmsExpand the following expression:

32y4x log

Start by applying property 1 to separate the product:

32y4x log 4 log 2 xlog 3y log

Properties of LogarithmsExpand the following expression:

Eliminate the exponents

32y4x log 4 log 2 xlog 3y log

4 log xlog2 y log3

Properties of LogarithmsRewrite the following logarithm:

75 -3x ln

For problems involving square roots, begin by converting the square root to a power

75 -3x ln

75 x 3ln

21

Properties of Logarithms

7

5 x 3ln 21

Apply property 1 to get rid of the quotient:

7

5 x 3ln 21

21

5) -(3x ln 7ln

Properties of Logarithms

21

5) -(3x ln 7ln

Apply property 3 to get rid of the exponent

21

5) -(3x ln 7ln 5) -(3x ln 21

7ln

Properties of LogarithmsRewrite the following logarithmic

expressions:

2y5 x ln

zy2xln 23 zln y 2ln 3ln x 2ln

y2ln 5) (x ln 21

Properties of LogarithmsExpand the following expression:

2) (x xln 2

212 ] 2) (x [xln

21

2) (x ln x

Properties of Logarithms2

12) (x ln x

21

2) (x ln ln x

2) (x ln 21 ln x

Properties of LogarithmsSection 3.3

Properties of LogarithmsSo far in this section, we have:

a) Change-of-Base Formulab) 3 Propertiesc) Expanded expressionsToday we are going to do the exact opposited) Condense expressions

Properties of LogarithmsWhen we were expanding, what order did

we typically apply the properties in?◦ Property 1 or Property 2◦End with Property 3

When we condense, we use the opposite order◦Property 3◦Property 1 or Property 2

Properties of LogarithmsThe most common error:

Log x – Log y

When you condense, you are condensing the expression down to one log function

y Log xLog

yx Log

Properties of LogarithmsCondense the following expression:

1) (x log 3 x log 21

Start by applying property 3, then move on to properties 1 and 2

321

1) (x log xlog

Properties of Logarithms32

1

1) (x log xlog

31) (x log x log

31) (x x log

Is this expression simplified to one log function?

Properties of LogarithmsCondense the following expression:

1)] (x log x [log 31

22

31

22 1)] (x log x log[

31

2 1)] x(x log[

32 1) x(x log

Properties of Logarithms

ln x )2 (x ln 2

2)] (zln y 4ln [2ln x 31

zln 21 y log3 x log 2

x

2) (x ln 2

y

zx log 3

2

2) (zy

xln 3 4

2

Properties of Logarithmsln x )2 (x ln 2

Properties of Logarithms zln

21 y log3 x log 2

Properties of Logarithms2)] (zln y 4ln [2ln x

31

Properties of Logarithms

Properties of Logarithms

Properties of Logarithms

Properties of Logarithms

Properties of Logarithms