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Alg2H 6-10, 6-11 Properties of Logarithms Class Lesson F Date _____

Properties of Exponents:

1. Product of two powers with equal bases: xa xb =

2. Quotient of two powers with equal bases:

3. Power of a power: (xa)b =

4. Power of a product: (xy)a =

5. Power of a quotient:

Definition of Logarithm:

Exponential Form: Logarithmic Form

bx = a _______________ where x>0, b>0, b 1

Properties of Logarithms:

1. logxxa = ____ 3. logbb = _____ 5. logb0 =_____

2. = _______ 4. logb1 = ______ 6. If logbx = logby, then _____ = _____

Since logarithms are exponents, there are other properties of logarithms that are

similar to the properties of exponents.

7. Logarithm of a Product

a) logb (xy) = ___________________________________________

Words: “The log of a product equals the _____________of the logs of the two factors”

b) Check with an example: log (3 5) = ______________________

_________ = _______________________

(Use calculator to find value of each side of the equation to verify that they are equal. Be sure to close each parenthesis on calculator.)

Try the proof of this property tonight for extra credit!!!

c) BE CAREFUL: Does (log3)(log5) = log 3 + log 5? (Check with calculator)

_________ = ____________

8. Logarithm of a Quotient

a) logb = _______________________________________

Words: “The log of a quotient equals the log of the numerator _______log of denominator”

b) Test with = ______________________________________________

______ = _______________________________________________

c) BE CAREFUL: Does = log 30 – log 5? ______________

x

y

FHGIKJ

log30

5

FHGIKJ

log

log

30

5

x

y

aFHGIKJ

b b xlog

x

x

a

b

Lesson F

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9. Logarithm of a Power:

a) logb(xn) = ________________________

Words: “The log of a power equals the exponent ________ the log of the base.”

b) Check with log 25 = _______________

_______ = _______________

c) BE CAREFUL: (log 2)5 5log2

Problems using Properties of Logarithms

II. Use the properties of logarithms to write each expression as the sum and/or

difference of logaX, logaY, logaZ with no exponents

1)

2)

III. Use the properties of logarithms to write each expression in terms of

c, d, e where c = logx2, d = logx5, e = logx7

3) 4)

5)

loga

X

YZ

2FHGIKJ

52

loga

Z

a

FHGIKJ

logx 35x log xx

20

7 3

FHGIKJ

log x x503

Lesson F

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IV. Use properties of logarithms to find the value of the given logarithm without using a calculator.

if log3 0.477 log5 0.699 log11 1.041

6) log 1500 7)

V. Use the properties of logarithms to write each expression as a single logarithm of a single

argument with coefficient 1

8) 9)

VI. Mixed Review

10) Simplify: 11) Simplify:

2 81

24 67 7 7log log log

log25000

11

FHGIKJ

1

3125 2 5 2log log log

3 641

2

21

2

2

3

x xFHGIKJ FHGIKJ

3 641

2

21

2

2

3

x xFHGIKJ FHGIKJ

7 9

7 27

34 231

36 320

x

x

c hc h

Lesson F

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Solve without calculator:

12) 13)

Solve without calculator: 14) (Hint: Evaluate inside parenthesis first)

log 1

2

128 x logx 83

2

log log log

64 2

163 3c he j x

Check your answers:

3) e + d + 1 4) d + 2c – e – 3 5) 2/3d + 1/3c +1/3 6) 3.176 7) 3.357

8) log7192 9) -1 10) x(-2/3)/144 or 1/144x(2/3) 11) 9x2/49

12) –7 13) ¼ 14) 1/3

Lesson F

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