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1 of 21
Pre-Calculus Chapter 3.3 – 3.4
Warm - up
3
45
2
11610
4
144
3
49
18 .12
25 .11
53 .10
08 .9
xz
yx
ba
completelySimplify
3 102 .9
15
1 .10
655 .11 ba
2
22
7
23 .12
z
zyx
2 of 21
Pre-Calculus Chapter 3.3 – 3.4
Chapter 3 Sec 3
Properties of Logarithms
3 of 21
Pre-Calculus Chapter 3.3 – 3.4
Essential Question
How do you rewrite logarithmic expressions to simplify or
evaluate them?
Key Vocabulary:Change-of-base formula
4 of 21
Pre-Calculus Chapter 3.3 – 3.4
Change of Base Formula• This allows you to write equivalent logarithmic
expressions that have different bases. For example change base 3 into base 10
a
xx
b
b
ba log
loglog
Base
a
xx
e
a ln
lnlog
Base
a
xxa
10
10
log
loglog
10 Base
5 of 21
Pre-Calculus Chapter 3.3 – 3.4
Change of BaseChange bases using common logarithms. Then approximate its value.
4log
25log25log a.
10
104
3219.26021.
3980.1
3log
18log18log b.
10
103
6309.24771.
2553.1
Change bases using natural logarithms. Then approximate its value.
4ln
25ln25log c. 4
3219.238629.1
21888.3
2ln
12ln12log d. 2
585.3693147.
48491.2
Example 1
6 of 21
Pre-Calculus Chapter 3.3 – 3.4
Properties 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.
1. Product Property:
2. Quotient Property:
3. Power Property:
vuvuuv aaa lnlnuvln logloglog
vuv
uvu
v
uaaa lnlnln logloglog
unuunu na
na lnln loglog
7 of 21
Pre-Calculus Chapter 3.3 – 3.4
Example 2Write each logarithm in terms of ln 2 and ln 3.
6ln a. 32ln 27
2ln b. 27ln2ln
5
17log c. 5
7
3ln2ln 33ln2ln 3ln32ln
Use properties of logarithms to verify:
5
1
75
7 7log7log 7log5
17
15
11
7log
7log7log 7
5
1
8 of 21
Pre-Calculus Chapter 3.3 – 3.4
Example 3Expand each logarithmic expression.
yx 23log a.8
14ln b.
x
yx loglog3log 2 8ln14ln 2
1
x
yx loglog23log 8ln14ln2
1 x
9 of 21
Pre-Calculus Chapter 3.3 – 3.4
Example 4Condense each logarithmic expression.
3log5log3
1 a. xx xx ln24ln4 b.
2loglog5
1 c. 33 xx
53 3loglog xx 24 ln4ln xx
2
44ln
x
x
53 3log xx
2log5
1 3 xx
53 2log xx
10 of 21
Pre-Calculus Chapter 3.3 – 3.4
Essential Question
How do you rewrite logarithmic expressions to simplify or evaluate them?
11 of 21
Pre-Calculus Chapter 3.3 – 3.4
Chapter 3 Sec 4
Solving Exponential and Logarithmic Equations
Essential Question
How do you solve exponential and logarithmic equations?
12 of 21
Pre-Calculus Chapter 3.3 – 3.4
Solving Equations• One-to-One
ax = ay if and only if x = yloga x = loga y if and only if x = y
• Inverse Propertiesxa xa log
xa xa log
13 of 21
Pre-Calculus Chapter 3.3 – 3.4
a. 2x = 32
b. ln x – ln 3 = 0
c.
d. ex = 7
e. ln x = –3
f. log x = –1
One-to-One
One-to-One
One-to-One
Inverse
Inverse
Inverse
93
1
x
Example 1
Original Equation
RewrittenEquation Solution
2x = 25
ln x = ln 3
3–x = 32
ln ex = ln 7
eln x = e–3
10log x = 10–1
x = 5
x = 3
x = –2
x = 7
x = e–3
x = 10–1 = 0.1
Property
14 of 21
Pre-Calculus Chapter 3.3 – 3.4
Solve each equation.
a. ex = 72ln ex = ln 72x = ln 72 ~ 4.28
Example 2
b. 3(2x) = 42 2x = 14 log22x = log214x = log21481.3
2ln
14lnx
234 c. 2 xe54 2 xe
4
52 xe4
5lnln 2 xe
4
5ln2 x 11.0
4
5ln
2
1x
15 of 21
Pre-Calculus Chapter 3.3 – 3.4
Solve the equation.Example 3
11432 52 t
1532 52 t
2
153 52 t
2
15log3log 3
523 t
5.7log52 3t
5.7log52 3t
42.35.7log2
1
2
53 t
16 of 21
Pre-Calculus Chapter 3.3 – 3.4
Solve the equation using quadratics.Example 4
0232 xx ee
0232
xx ee
012 xx ee
02 xe 01 xe
2xe 1xe1lnx
0x
69.02ln x
17 of 21
Pre-Calculus Chapter 3.3 – 3.4
Solve each equation.
a. ln 3x = 2eln 3x = e2 3x = e2
Example 5
4ln25 c. x
46.23
1 2 ex
7log15log b. 33 xx
715 xx
2
84
x
x
1ln2 x
2
1ln x
2
1ln
ee x 61.02
1
ex
43log2 d. 5 x
23log 5 x23log 55 5 x
3
25
253
x
x
18 of 21
Pre-Calculus Chapter 3.3 – 3.4
Solve the equation check for extraneous solutions.
Example 6
xxx ln232ln2ln
2ln322ln xxx
22 ln672ln xxx 22 672 xxx
0672 xx
016 xx
101
606
xx
xx
Check:
36ln36ln
6ln29ln4ln
6ln2312ln26ln
6
x
1ln21ln1ln
1ln232ln21ln
1
x
ln (–1) is invalid.
X
19 of 21
Pre-Calculus Chapter 3.3 – 3.4
Solve each equation.
You have deposited $500 in an account that pays 6.75% interest, compounded continuously. How long will it take your money to double?
Example 5
te 0675.5001000 tr t ePeA 0 675.500te 0675.2
te 0675.ln2ln t0675.2ln
27.100675.
2lnt
20 of 21
Pre-Calculus Chapter 3.3 – 3.4
Essential Question
How do you solve exponential and
logarithmic equations?
21 of 21
Pre-Calculus Chapter 3.3 – 3.4
Daily Assignment• Chapter 3.3 -3.4• Text Book
• Pgs 211 – 212 • #1 – 45 and 59 – 73 Mode 4 (1,5,9,13…73)
• Pgs 221 – 222 • #1 – 21 Mode 4; #29 – 49 Mode 4;
#85 – 97 Mode 4• Read Section 3.5
22 of 21
Pre-Calculus Chapter 3.3 – 3.4
Ch 3aPop Quiz
3 Find
123For .3
9
3Simplify .2
276624
GCF) find :(hint completelyFactor .1
2
322
2
23
xf
xxxf
zyx
xyz
xxx