Math 1310
Chapter 5 – Section 5.3
Logarithms
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Guidelines and tips
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Logarithmic Functions: Properties and Graphs, Study notes of Algebra
The basics of logarithmic functions, including their definition, properties, and graphs. Topics include writing equations in exponential and logarithmic form, inverse properties, and characteristics of logarithmic functions. Numerous examples are provided for clarification.
Typology: Study notes
Pre 2010
1 / 11
This page cannot be seen from the preview
Don't miss anything!
Related documents
Partial preview of the text
Download Logarithmic Functions: Properties and Graphs and more Study notes Algebra in PDF only on Docsity!
Math 1310
Chapter 5 – Section 5.
Logarithms
Logarithmic Function
with base
b
For
b^
0 and
b^ not equal to 1, the
logarithmic function with base
b , denoted by
is
given byThus,
is the exponent in which the base
b^ must be raised to give
x.
The logarithmic function with base 10 is called the
common logarithmic function
and is
denoted byThe logarithmic function with base
e^ is called the
natural logarithmic function
and is denoted by
if
only
and
if
log
x
b
y
x^
y
b^
log
xb
xb
log
log
x
xf
ln
x
xf
Example 5: Write the following equation in logarithmic form.Example 6: Simplify the following expression.Example 7: Simplify the following expression.Example 8: Simplify the following expression.Example 9: Simplify the following expression.
x
e^
4 =^9
log
3 1 log
9
121 (^1111) log
log
2
Example 10: Simplify the following expression.
Inverse Properties of Logarithms
For
b^
0 and
b^ not equal to 1,
log
x
xb b
log
x
b^
xb = log
Example 16: Find the value of
x^ in the following equation.
Example 17: Find the value of
x^ in the following equation. Characteristics of Logarithmic Functions
of the form
•^
The graphs of all logarithmic functions of the form
pass through the key point
•^
The
y -axis (
x^ = 0) is the vertical asymptote.
•^
The domain is
and the range is all real numbers.
3
log
4
= x
1 2 7 log
= x
x
x
f^
b
log
(^
x
x
f^
b
log
(^
) , (^0) ( ∞
If
b
> 1 then the graph of
looks
like:
y
x
x
x
f^
b
log
(^
Example 18: Apply transformations to sketch the graph of the following function. Show theasymptote and then state the domain and range of the function.Example 19: Apply transformations to sketch the graph of the following function. Show theasymptote and then state the domain and range of the function.
1 ) 2 ln( )(
=^
x xf
log( )(
x
xf
Example 20: Find the domain of following function algebraically.Example 21: Find the domain of following function algebraically.Example 22: Find the domain of following function algebraically.
ln(
=^
x
xf
log
2 3
=^
x
xf
log
2 2