Quiz 3.2

• Write the log as an exponent. • Evaluate the log without using your calculator. 4

3.3Properties of Logarithms

Date: ____________

Change of Base Theorem

logax = log x log a

logax = ln x ln a

OR

1) log4 9

2) log7 3

Write the logarithm in terms of

common logarithms.

log 9

log 4

log 3

log 7

1) log5 8

2) log2 7

Write the logarithm in terms of

natural logarithms.

ln 8

ln 5

ln 7

ln 2

Evaluating Logarithms

log523 = log23 log5 ≈ 1.948

OR

log523 = ln23 ln5 ≈ 1.948

Other Examples

log3149 = log149 log3 ≈ 4.555

log7300 = log300 log7 ≈ 2.9312

Properties of Logarithms

Product Property

Quotient Property

Power Property

Let b, u, and v be positive numbers such that b ≠ 1.

logbuv = logbu + logbv

logbun = nlogbu

logb = logbu − logbvuv

1) log9

4

7

2) log9 28

Use log9 7 0.8856 and log9 4 0.6309

to approximate the following.

0.2547

0.6309 0.8856 1.5165

3)log9 256 4 0.6309 2.5236

log9 4 log9 7 0.6309 0.8856

9log (4 7) log9 4 log9 7

49log 4 4log9 4

1) log3

2

7

2) log314

Use log32 0.6310 and log3 7 1.7712

to approximate the following.

1.1402

0.6310 1.7712 2.4022

3) log3128 7 0.6310 4.4170

log32 log3 7 0.6310 1.7712

3log (2 7) log32 log3 7

73log 2 7log32

Find the exact value of the logarithm.

48log 8

14

8log 81

4

4 3ln e34ln e

3

4

31

log81

13log 81 31 log 81

43log 3 4

Use the properties of logarithms to simplify the given expression.

51

log15

5 5log 1 log 15

50 log 15

5 5log 5 log 3

51 log 3

51 log 3

5log (5 3)

Use the properties of logarithms to simplify the given expression.

31

log18

3 3log 1 log 18

30 log 18

3 3log 9 log 2 2

3 3log 3 log 2 32 log 2

3log 9 2

Use the properties of logarithms to simplify the given expression.

3log 3012

3log (30)1

32log 30

13 32

log 3 log 10

132

1 log 10 1

32log 10

132

log 3 10

Write as the sum, difference, or product of logarithms. Simplify, if possible.

log4

4x6

y = log44x6 – log4 y

= log44 + log4x6 – log4 y

= log44 + 6log4x – log4 y

= 1 + 6log4x – log4 y

Write as the sum, difference, or product of logarithms. Simplify, if possible.

log7

6x3y = log76

– log7 x3y

= log76 −(log7x3 + log7 y)

= log76 − (3log7x + log7 y)

= log76 − 3log7x − log7 y

Write as the sum, difference, or product of logarithms. Simplify, if possible.

log57 x

= log57 + log5 x

= log57 + log5x½

= log57 + ½log5x

Write as the sum, difference, or product of logarithms. Simplify, if possible.

loga

x3y5

z = loga

x3y5

z

½

= ½loga

x3y5

z

= ½(loga x3+ loga y

5 – logaz)

= ½(3loga x+ 5loga y – logaz)

Condense the expression to the logarithm of a single quantity.

log35 + 6log3x − log3 7

= log35 + log3x6 − log3 7

= log3(5x6) − log3 7

= log3 7 5x6

Condense the expression to the logarithm of a single quantity.

3log8 x − 5log8 y + log8 15

= log8x3 − log8 y

5 + log8 15

•15 log8y5

= x3

= log8y5

15x3

x3

Condense the expression to the logarithm of a single quantity.

3log8 x − 5log8 y − log8 15

= log8x3 − log8 y

5 − log8 15

= log815y5