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4 minutes. Warm-Up. Solve each equation for x. Round your answers to the nearest hundredth. 1) 10 x = 1.498. 2) 10 x = 0.0054. Find the value of x in each equation. 3) x = log 4 1. 4) ½ = log 9 x. 6.4.1 Properties of Logarithmic Functions. Objectives:

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Warm-Up

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Warm up

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

6.4.1 Properties of Logarithmic Functions

Objectives:

Simplify and evaluate expressions involving logarithms

Solve equations involving logarithms


Properties of logarithms

Properties of Logarithms

For m > 0, n > 0, b > 0, and b  1:

Product Property

logb (mn) = logb m + logb n


Example 1

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 logarithms1

logb = logb m – logb n

m

n

Properties of Logarithms

For m > 0, n > 0, b > 0, and b  1:

Quotient Property


Example 2

12

= log5

10

Example 2

given: log5 12  1.5440

log5 10  1.4307

log5 1.2

= log5 12 – log5 10

1.5440 – 1.4307

0.1133


Properties of logarithms2

Properties of Logarithms

For m > 0, n > 0, b > 0, and any real number p:

Power Property

logb mp = p logb m


Example 3

Example 3

given: log5 12  1.5440

log5 10  1.4307

log5 1254

5x = 125

= 4 log5 125

53 = 125

=4  3

x = 3

= 12


Practice

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

Homework

p.382 #13-21 odds,31,35


Warm up1

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

6.4.2 Properties of Logarithmic Functions

Objectives:

Simplify and evaluate expressions involving logarithms

Solve equations involving logarithms


Properties of logarithms3

Properties of Logarithms

For b > 0 and b  1:

Exponential-Logarithmic Inverse Property

logb bx = x

and b logbx = x for x > 0


Example 11

Example 1

Evaluate each expression.

a)

b)


Practice1

Practice

Evaluate each expression.

1) 7log711 – log3 81

2) log8 85 + 3log38


Properties of logarithms4

Properties of Logarithms

For b > 0 and b  1:

One-to-One Property of Logarithms

If logb x = logb y, then x = y


Example 21

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


Practice2

Practice

Solve for x.

1) log5 (3x2 – 1) = log5 2x

2) logb (x2 – 2) + 2 logb 6 = logb 6x


Homework1

Homework

p.382 #29,33,37,43,47,49,51,57,59,61


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