Download Use the Four Basic Properties of Logarithms Use the Product Rule

Math 35 (Spring ’09)
9.7 "Properties of Logarithms"
Objectives:
*
Use the properties of logarithms
*
Write logarithmic expressions as a single logarithm
*
Use the change-of-base formula
Use the Four Basic Properties of Logarithms
The Natural Exponential Function:
The function de…ned by
is the natural exponential function where e = 2:71828::::
The Natural Logarithmic Function:
The natural logarithmic function with base e is de…ned by the equation:
where
Properties of Logarithms:
For all positive numbers b, where b 6= 1; and x > 0:
1:
2:
3:
4:
Example 1: (Using the four basic properties of logarithms)
Simplify each expression.
a)
log4 4 =
b)
log2 24 =
c)
ln e3 =
d)
5log5 2 =
Use the Product Rule for Logarithms
The Product Rule for Logarithms:
For all positive real numbers M; N; and b, where b 6= 1;
In words, "the logarithm of a product is equal to the sum of the logarithms"
Example 2: (Using the product rule)
Write each expression as a sum of logarithms. Then simplify, if possible.
a)
log3 (3 4) =
b)
log5 25xy =
Use the Quotient Rule for Logarithms
The Quotient Rule for Logarithms:
For all positive real numbers M; N; and b, where b 6= 1;
In words, "the logarithm of a quotient is equal to the di¤ erence of the logarithms"
Example 3: (Using the quotient rule)
Write each expression as a di¤ erence of logarithms. Then simplify, if possible.
exy
a)
log6 56 =
b)
ln
=
z
Page: 1
Notes by Bibiana Lopez
Intermediate Algebra by Tussy and Gustafson
9.7
Use the Power Rule for Logarithms
The Power Rule for Logarithms:
For all positive real numbers M; and b, where b 6= 1; and any real number p;
In words, "the logarithm of a power is equal to the power times the logarithm"
Example 4: (Using the power rule)
Write each logarithm without an exponent or a square root.
a)
ln x4 =
b)
log
p
10 =
Example 5: (Using the properties of logarithms)
Write each logarithm as the sum/or di¤ erence of logarithms of a single quantity.
r
3
4 x y
2 3
a)
logb x y z
b)
log3
z
Write Logarithmic Expressions as a Single Logarithm
Example 6: (Writing logarithmic expressions as a single logarithm)
Write each logarithmic expression as one logarithm.
p
a)
log5 x3 + log5 y 1=2
b)
log2 x
2
log2 y + 3 log2 z
In summary,
If b, M; and N are positive real numbers, b 6= 1; and p is any real number.
1: logb 1 = 0
5: logb M N = logb M + logb N
M
= logb M logb N
2: logb b = 1
6: logb
N
x
p
3: logb b = x
7: logb M = p logb M
logb x
4: b
=x
Page: 2
Notes by Bibiana Lopez