8.5 Properties of Logarithms

Objectives:1. Compare & recall the properties of

exponents2. Deduce the properties of logarithms

from/by comparing the properties of exponents

3. Use the properties of logarithms 4. Application

Vocabulary:change-of-base formula

Pre-Knowledge

For any b, c, u, v R+, and b ≠ 1, c ≠ 1, there exists some x, y R, such that

u = bx, v = by

By the previous section knowledge, as long as taking

x = logbu, y = logbv

1. Product of Power

am an = am+n

1. Product Property

logbuv = logbu + logbv

Proof

logbuv = logb(bxby)= logbbx+y = x + y

= logbu + logbv

2. Quotient Property

2. Quotient of Power

a

aa

m

nm n

vlogulogv

ulog bbb

Proof

vlogulogyxblogb

blog

v

ulog bb

y xby

x

bb

3. Power of Power

(am)n = amn

3. Power Property

logbut = t logbu

Proof

logbut = logb(bx)t = logbbtx = tx = t logbu

3. Power of Power

(am)n = amn

3. Power Property

logbut = t logbu

4. Change-of-Base Formula

blog

ulog ulog

c

cb

Proof Note that

bx = u, logbu = x

Taking the logarithm with base c at both sides:

logcbx = logcu or x logcb = logcu

blog

ulog ulog

c

cb

blog

1

ulog

blog

1

blog

ulog ulog more, Further

u

c

cc

cb

Example 1 Assume that log95 = a, log911 = b, evaluate

a) log9 (5/11)

b) log955

c) log9125

d) log9(121/45)

Practice

A) P. 496 Q 9 – 10 by assuming log27 = a, and

log221 = b

B) P. 496 Q 14 – 17

Example 2 Expanding the expression

a) ln(3y4/x3)ln(3y4/x3) = ln(3y4) – lnx3 = ln3 + lny4 – lnx3

= ln3 + 4 ln|y|– 3 lnx

b) log3125/6x9

log3125/6x9 = log3125/6 + log3x9

= 5/6 log312 + 9 log3x

= 5/6 log3(3· 22) + 9 log3x

= 5/6 (log33 + log322) + 9 log3x

= 5/6 ( 1 + 2 log32) + 9 log3x

Practice Expand the expression

P. 496 Q 39, 45

Example 3 Condensing the expressiona) 3 ( ln3 – lnx ) + ( lnx – ln9 )

3 ( ln3 – lnx ) + ( lnx – ln9 ) = 3 ln3 – 3 lnx + lnx – 2 ln3 = ln3 – 2 lnx = ln(3/x2)

b) 2 log37 – 5 log3x + 6 log9y2

2 log37 – 5 log3x + 6 log9y2

= log349 – log3x5 + 6 ( log3y2/ log39)

= log3(49/x5) + 3 log3y2

= log3(49y6/x5)

Practice Condense the expression

P. 497 Q 56 - 57

Example 4 Calculate log48 and log615 using common and natural logarithms.

a) log48

log48 = log8 / log4 = 3 log2 / (2 log2)= 3/2 log48 = ln8 / ln4 = 3 ln2 / (2 ln2) = 3/2

b) log615 = log15 / log6 = 1.511

Example 5 The Richter magnitude M of an earthquake is based on the intensity I of the earthquake and the intensity Io of an earthquake that can be barely felt. One formula used is M = log(I / Io). If the intensity of the Los Angeles earthquake in 1994 was 106.8 times Io, what was the magnitude of the earthquake? What magnitude on the Richter scale does an earthquake have if its intensity is 100 times the intensity of a barely felt earthquake?

I / Io = 106.8, M = log(I / Io) = log106.8 = 6.8

I / Io = 100, M = log(I / Io) = log100 = 2

Challenge Simplify (No calculator)

1)

2)

3)

4)

5) Proof

)3(2log32

)5353log(

9106log10)(log 32

3

xddccbbaa loglogloglog

2log

1

log

1

52

ππ

Assignment:

8.4 P496 #14-52 - Show work

8.5 Properties of Logarithmic