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

Math 35
9.7 "Properties of Logarithms"
Objectives:
* Use the properties of logarithms.
* Write logarithmic expressions as single logarithms.
* Use the change-of-base formula.
Use the Four Basic Properties of Logarithms
The Natural Exponential Function:
The function de…ned by f (x) = ex 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 f (x) = ln x where ln x = loge x
:
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 r
quantity.
x3 y
4
a) logb x2 y 3 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.
a) log5 x3 + log5 y 1=2
b) log2
p
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
2: logb b = 1
3: logb bx = x
5: logb M N = logb M + logb N
M
6: logb
= logb M logb N
N
7: logb M p = p logb M
4: blogb x = x
Page: 2
Notes by Bibiana Lopez