Properties of Logarithmic Functions Properties of Logarithmic Functions Objectives: Simplify and...
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Transcript of Properties of Logarithmic Functions Properties of Logarithmic Functions Objectives: Simplify and...
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