Properties of Logarithmic FunctionsProperties of Logarithmic FunctionsProperties of Logarithmic FunctionsProperties of Logarithmic Functions

Objectives: •Simplify and evaluate expressions involving logarithms•Solve equations involving logarithms

Properties of Logarithms

Product Propertylogb (mn) = logb m + logb n

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

Example 1

= log5 12 + log5 10

log5 120 =

given: log5 12 1.5440log5 10 1.4307

log5 (12)(10)

1.5440 + 1.4307 2.9747

Properties of Logarithms

Quotient Property

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

logb = logb m – logb nm

n

Example 2given: log5 12 1.5440log5 10 1.4307

log5 1.2

= log5 12 – log5 10 1.5440 – 1.4307 0.1133

= log5 12

10

Properties of Logarithms

Power Property

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

logb mp = p logb m

Example 3given: log5 12 1.5440log5 10 1.4307

log5 1254 = 4 3

= 12

= 4 log5 125

5x = 12553 = 125x = 3

PracticeWrite 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

Warm-UpWrite each expression as a single logarithm. Then simplify, if possible.

4 minutes

1) log6 6 + log6 30 – log6 5

2) log6 5x + 3(log6 x – log6 y)

Properties of Logarithms

Exponential-Logarithmic Inverse Propertylogb bx = x

For b > 0 and b 1:

and b logb

x = x for x > 0

Example 1Evaluate each expression.

5log 347log 7 5

5log 374log 7 5

4 3 7

5log 34 5

9log 24

19 log

4

4

12 log

4

4 42 log 1 log 4

2 (0 1) 3

a)

b)

PracticeEvaluate each expression.1) 7log

711 – log3 81

2) log8 85 + 3log3

8

Properties of Logarithms

One-to-One Property of LogarithmsIf logb x = logb y, then x = y

For b > 0 and b 1:

Example 2Solve log2(2x2 + 8x – 11) = log2(2x + 9) for x.

log2(2x2 + 8x – 11) = log2(2x + 9) 2x2 + 8x – 11 = 2x + 9

2x2 + 6x – 20 = 02(x2 + 3x – 10) =

02(x – 2)(x + 5) = 0 x = -

5,2Check:

log2(2x2 + 8x – 11) = log2(2x + 9)log2 (–1) = log2 (-1)

undefinedlog2 13 = log2 13 true

PracticeSolve for x.

1) log5 (3x2 – 1) = log5 2x

2) logb (x2 – 2) + 2 logb 6 = logb 6x

Solving Equations and ModelingSolving Equations and ModelingSolving Equations and ModelingSolving Equations and ModelingObjectives: •Solve logarithmic and exponential equations by using algebra and graphs•Model and solve real-world problems involving logarithmic and exponential relationships

Summary of Exponential-Logarithmic Definitions and

Properties

Definition of logarithm y = logb x only if by = x

Product Property logb mn = logb m + logb n

Quotient Property

Power Property

logb mp = p logb m

logb = logb m – logb n

mn( )

Summary of Exponential-Logarithmic Definitions and

Properties

Exp-Log Inverse b logb x = x for x > 0 logb bx = x for all x

1-to-1 for Exponents

bx = by; x = y

1-to-1 for Logarithms

logb x = logb y; x = y

Change-of-Base logc a =logb a

logb c

Example 1

Solve for x.3x – 2 = 4x + 1

log 3x – 2 = log 4x + 1

(x – 2) log 3 = (x + 1) log 4x log 3 – 2 log 3 = x log 4 +

log 4x log 3 – x log 4 = log 4 + 2 log 3x (log 3 – log 4) = log 4 + 2 log 3

x –12.46

log 4 + 2 log 3log 3 – log 4

=x

Example 2

Solve for x.

log x + log (x + 3) = 1log [x(x + 3)] =

1101 = x(x + 3)

x2 + 3x – 10 = 0

(x + 5)(x – 2) = 0

x = 2,-5

101 = x2 + 3x

Example 2

Solve for x.

log x + log (x + 3) = 1

log x + log (x + 3) = 1

x = 2,-5

Let x = 2

log 2 + log (2 + 3) = 1 log 2 + log 5 =

1 1 = 1

log x + log (x + 3) = 1

Let x = -5

log -5 + log (-5 + 3) = 1 log -5 + log -2

= 1 undefined

x = 2

Check:

Example 3

Solve for x.

8e2x-5 = 56e2x-5 = 7ln e2x-5 = ln

72x - 5 = ln 7

x =

ln 7 + 52

x 3.47

Example 4Suppose that the magnitude, M, of an earthquake measures 7.5 on the Richter scale. Use the formula below to find the amount of energy, E, released by this earthquake.

11.8

2log3 10

EM

11.811.25 log

10

E

11.8

27.5 log

3 10

E

11.2511.8

1010

E

11.25 11.810 10 E 231.12 10 E 231.12 10 ergs