4 minutes
Warm-Up
Solve each equation for x. Round your answers to the
nearest hundredth.
1) 10x = 1.498
2) 10x = 0.0054
Find the value of x in each equation.
3) x = log4 1
4) ½ = log9 x
6.4.1 Properties of Logarithmic Functions
Objectives:
•Simplify and evaluate expressions involving logarithms
•Solve equations involving logarithms
Properties of Logarithms
For m > 0, n > 0, b > 0, and b  1:
Product Property
logb (mn) = logb m + logb n
Example 1
given: log5 12  1.5440
log5 10  1.4307
log5 120 = log5 (12)(10)
= log5 12 + log5 10
 1.5440 + 1.4307
 2.9747
Properties of Logarithms
For m > 0, n > 0, b > 0, and b  1:
Quotient Property
m
logb
= logb m – logb n
n
Example 2
given: log5 12  1.5440
log5 10  1.4307
12
log5 1.2 = log5
10
= log5 12 – log5 10
 1.5440 – 1.4307
 0.1133
Properties of Logarithms
For m > 0, n > 0, b > 0, and any real number p:
Power Property
logb mp = p logb m
Example 3
given: log5 12  1.5440
log5 10  1.4307
log5 1254 = 4 log5 125
= 43
= 12
5x = 125
53 = 125
x=3
Practice
Write each expression as a single logarithm.
1) log2 14 – log2 7
2) log3 x + log3 4 – log3 2
3) 7 log3 y – 4 log3 x
Homework
p.382 #13-21 odds,31,35
4 minutes
Warm-Up
Write each expression as a single logarithm. Then
simplify, if possible.
1) log6 6 + log6 30 – log6 5
2) log6 5x + 3(log6 x – log6 y)
6.4.2 Properties of Logarithmic Functions
Objectives:
•Simplify and evaluate expressions involving logarithms
•Solve equations involving logarithms
Properties of Logarithms
For b > 0 and b  1:
Exponential-Logarithmic Inverse Property
logb bx = x
and
b logbx = x for x > 0
Example 1
Evaluate each expression.
a) log 7 7
5
log5 3
4
 4log 7 7  5
log5 3
 45
 43
7
log5 3
b)
9
log9 2
1
 log 4
4
1
 2  log 4
4
 2   log 4 1  log 4 4 
 2  (0  1)
3
Practice
Evaluate each expression.
1) 7log711 – log3 81
2) log8 85 + 3log38
Properties of Logarithms
For b > 0 and b  1:
One-to-One Property of Logarithms
If logb x = logb y, then x = y
Example 2
Solve log2(2x2 + 8x – 11) = log2(2x + 9) for x.
log2(2x2 + 8x – 11) = log2(2x + 9)
2x2 + 8x – 11 = 2x + 9
2x2 + 6x – 20 = 0
2(x2 + 3x – 10) = 0
2(x – 2)(x + 5) = 0
x = -5,2
Check: log2(2x2 + 8x – 11) = log2(2x + 9)
log2 (–1) = log2 (-1) undefined
log2 13 = log2 13
true
Solve for x.
Practice
1) log5 (3x2 – 1) = log5 2x
2) logb (x2 – 2) + 2 logb 6 = logb 6x
Homework
p.382 #29,33,37,43,47,49,51,57,59,61