# 8 5 Properties of Logarithms

Document Sample

```					  8.5 Properties of Logarithms

Objectives:
1. Compare & recall the properties of
exponents
2. 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 of Power
m
a      m n
n
a
a
2. Quotient Property
u
logb  logbu  logb v
v
Proof
u       bx
log b  log b y  log b b x y
 x  y  log b u  log b v
v       b
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
log c u
log b u 
log c b
Proof   Note that
bx = u,      logbu = x
Taking the logarithm with base c at both sides:
logcbx = logcu        or          x logcb = logcu
log c u
logb u 
log c b
log cu     1      1
Further more, log bu                 
log cb   log cb log ub
log cu
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 expression
a) 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)        loga blogb clogc d logd x
a
2)     (log3 10)2  6log3 10  9

3)   log2 3 (2  3 )

4)   log( 3  5  3  5 )

5)   Proof        1

1
2
log 2 π log 5 π
8.5 Properties of Logarithmic

Assignment:

8.4 P496 #14-52 - Show work

```
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
 views: 35 posted: 5/30/2012 language: pages: 17
How are you planning on using Docstoc?