Alg2H
6-10, 6-11 Properties of Logarithms Class Lesson F Date _____
Lesson F
p.1
Properties of Exponents:
1. Product of two powers with equal bases: xa  xb =
2. Quotient of two powers with equal bases:
xa

xb
3. Power of a power: (xa)b =
4. Power of a product: (xy)a =
5. Power of a quotient:
FG x IJ
H yK
Definition of Logarithm:
Exponential Form:
bx = a
a


Logarithmic Form
_______________
where x>0, b>0, b 1
Properties of Logarithms:
1. logxxa = ____
3. logbb = _____
2. blogb x = _______ 4. logb1 = ______
5. logb0 =_____
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
x
a) logb
= _______________________________________
y
Words:
“The log of a quotient equals the log of the numerator _______log of denominator”
FG IJ
H K
b) Test with log
FG 30IJ
H 5K
= ______________________________________________
______ = _______________________________________________
c) BE CAREFUL: Does
log 30
= log 30 – log 5? ______________
log 5
1
9. Logarithm of a Power:
Lesson F
p.2
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
FG IJ
H K
2
1) log X 
a
YZ
2) 5 log a
F ZI 
GH a JK
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) log x 35x
4) log x
FG 20 IJ 
H 7x K
3
3
5) log x 50 x 
2
Lesson F
p.3
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
7) log
6) log 1500
FG 25000IJ
H 11 K
V. Use the properties of logarithms to write each expression as a single logarithm of a single
argument with coefficient 1
1
1
log 125  2 log 5  log 2 
8) 2 log 7 8  log 7 4  log 7 6 
9)
2
3
VI. Mixed Review
10) Simplify:
1 2
FG 3x IJ  FG 64 x IJ
H K H K
2
1
2
2
3
11) Simplify:
c h
 c27 x h
7 34  9 x 2
7 36
31
3 20
3
Solve without calculator:
13) log x 8  
12) log 1 128  x
2
Solve without calculator:
e c
hj
14) log log 3log3 16  x
64
2
3
2
Lesson F
p.4
(Hint: Evaluate inside parenthesis first)
Check your answers:
3) e + d + 1
4) d + 2c – e – 3 5) 2/3d + 1/3c +1/3 6) 3.176
8) log7192
9) -1
10) x(-2/3)/144 or 1/144x(2/3)
12) –7 13) ¼ 14) 1/3
7) 3.357
11) 9x2/49
4