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Warm-Up

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4 minutes

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

For m > 0, n > 0, b > 0, and b 1:

Product Property

logb (mn) = logb m + logb n

given: log5 12 1.5440

log5 10 1.4307

log5 120 =

log5 (12)(10)

= log5 12 + log5 10

1.5440 + 1.4307

2.9747

logb = logb m – logb n

m

n

For m > 0, n > 0, b > 0, and b 1:

Quotient Property

12

= log5

10

given: log5 12 1.5440

log5 10 1.4307

log5 1.2

= log5 12 – log5 10

1.5440 – 1.4307

0.1133

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

Power Property

logb mp = p logb m

given: log5 12 1.5440

log5 10 1.4307

log5 1254

5x = 125

= 4 log5 125

53 = 125

=4 3

x = 3

= 12

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

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

4 minutes

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

For b > 0 and b 1:

Exponential-Logarithmic Inverse Property

logb bx = x

and b logbx = x for x > 0

Evaluate each expression.

a)

b)

Evaluate each expression.

1) 7log711 – log3 81

2) log8 85 + 3log38

For b > 0 and b 1:

One-to-One Property of Logarithms

If logb x = logb y, then x = y

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.

1) log5 (3x2 – 1) = log5 2x

2) logb (x2 – 2) + 2 logb 6 = logb 6x

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