Download Math 102 5.3 "Logarithms" Objectives: * Switch between exponential

Math 102
5.3 "Logarithms"
Objectives:
* Switch between exponential and logarithmic form.
* Solve logarithmic equations.
* Apply the properties of logarithms to simplify expressions.
De…nition:
"Logarithm"
If r is any positive real number, then the unique exponent t such that bt = r is called the logarithm of r with base b
logb r = t is equivalent to bt = r
and is denoted by logb r
:
:
Example 1: (Switching between exponential and logarithmic form)
Write each equation in logarithmic form.
a) 25 = 32
b) 5
3
=
1
125
c)
3
2
3
=
27
8
Example 2: (Switching between exponential and logarithmic form)
Write each equation in exponential form.
a) log3 27 = 3
b) log5
1
25
=
2
c) log10 0:1 =
1
Some logarithms can be determined by changing to exponential form and using the properties of exponents, as in the
next example.
Example 3: (Evaluating logarithmic expressions)
Evaluate each expression.
a) log6 36
b) log4
1
64
c) log5
p
3
25
Some equations that involve logarithms can also be solved by changing them to exponential form and using our knowledge
of exponents.
Example 4: (Solving logarithmic equations)
Solve each equation.
a) logb
27
64
=3
b) log2=3 x = 2
c) log5 w = 2
Page: 1
Notes by Bibiana Lopez
College Algebra by Kaufmann and Schwitters
5.3
Properties of Logarithms
Product Rule for Logarithms:
For positive numbers b; r; and s; where b 6= 1;
:
Example 5: (Using the product rule for logarithms)
Given that log2 5 = 2:3219 and log2 7 = 2:8074, evaluate log2 35:
Quotient Rule for Logarithms:
For positive numbers b; r; and s; where b 6= 1;
:
Example 6: (Using the quotient rule for logarithms)
Given that log2 5 = 2:3219 and log2 7 = 2:8074, evaluate log2
7
5
:
Power Rule for Logarithms:
If r is a positive real number, b is a positive real number other than 1, and p is any real number,
then
:
Example 7: (Using the quotient rule for logarithms)
Given that log2 5 = 2:3219, evaluate log2 125:
Page: 2
Notes by Bibiana Lopez
College Algebra by Kaufmann and Schwitters
5.3
Properties of Logarithms:
If r; s, and b are positive real numbers with b 6= 1; and p is any real number, then
1:
5:
2:
6:
3:
7:
4:
8:
Example 8: (Using the properties of logarithms)
Rewrite each expression using sums or di¤ erences of multiples of logarithms.
r
2
x
a) logb xy
b) logb
y
c) logb
p
3
x2 z
d) logb
x3
y2 z
Example 9: (Using the properties of logarithms)
Rewrite each expression as a single logarithm.
a) 2 logb x
4 logb y
b) logb x + 5 logb y
Page: 3
Notes by Bibiana Lopez
College Algebra by Kaufmann and Schwitters
c) 2 logb x + 4 logb y
3 logb z
5.3
d) 2 logb x +
1
3
logb (x
1)
1
5
logb (3x + 2)
Example 10: (Solving logarithmic equations)
Solve each equation.
a) log7 5 + log7 x = 1
b) log10 x + log10 (x
c) log2 (x
d) log4 (x + 3) = 2
1)
log2 (x + 3) = 2
Page: 4
3) = 1
log4 7
Notes by Bibiana Lopez