Logarithmic Functions Do the following Find the

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Logarithmic Functions Do the following Find the inverse of f(x) Definition Mathematicians have defined a function, called logarithm that allows one to write the inverse of f(x) Definition of the Logarithmic Function For x 0 and b

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Logarithmic Functions

Do the following Find the inverse of f(x) =

Definition Mathematicians have defined a function, called ‘logarithm’ that allows one to write the inverse of f(x) =

Definition of the Logarithmic Function For x > 0 and b > 0, b ≠ 1
y = is equivalent to = x

The function f(x) = is the logarithmic function with base ‘b’

Forms Logarithmic form:

Exponential form:

Notice Logarithmic form: Where is the exponent and base.

Exponential form:

Base of the logarithm is the base of the exponent.

Evaluate Logarithms Rewrite in exponential form, then evaluate
Example:
Evaluate

Solution:1) Write as an equation: = x
2) Rewrite in exponential form:
3) Solve

Try b)

c)

Basic Logarithmic Properties 1) note:

2) note:

a)

b)

Domain of a Logarithmic Function Domain of f(x) = consists of all x for which g(x) > 0

Find the domain of g(x) =

Solution: domain consist of all x for which x + 3 > 0.
So, solve the inequality, obtain x > - 3 so the domain is

Common Logarithms Logarithmic function with base 10.
Function f(x) = is usually expressed as f(x) = log x.

Calculator with LOG key can be used to evaluate common logarithms.
Find:
log 1000
log 2.5
Error message or NONREAL ANSWERS, out of the domain

Properties of Common Logarithms General Properties
1)

2)

3)

4)

3 and 4 are inverse properties Common Logarithms
1) log 1 = 0

2) log 10 = 1

3)

4)

Natural Logarithms Logarithmic function with base ‘e’
Function f(x) = usually expressed as f(x) = ln x
ln x read ‘el en of x’

Calculator with LN key can be used to evaluate natural logarithms.

Domain: f(x) = ln x
all values of x > 0

Properties of Natural Logarithms General Properties
1)

2)

3)

4)

3 and 4 are inverse properties Natural Logarithms
1) ln 1 = 0

2) ln e = 1

3)

4)

Try

Summary y = is equivalent to = x
Logarithmic and exponential form
Evaluation of logarithms
Use definition
Calculator
Common and natural logarithms
Logarithm properties.

Inverse of f(x) = Step 1: Replace f(x) with y

Sep 2: Interchange x and y

Step 3: Solve for y

Step 4: Replace y with

Logarithmic Functions Do the following Find the

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