lesson f alg2h 6-10, 6-11 properties of logarithms class ... · 3 iv. use properties of logarithms...
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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
p.2
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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
p.3
4
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
p.4