1. Section 3.3a and b!!! Homework: p. 308 1-35 odd, 41-57 odd
Logarithms
Section 3.3a and b!!!
Homework: p odd, odd

2. What does the horizontal line test tell us???
First, remind me…
What does the horizontal line test tell us???
More specifically, what does it tell us about the function
This function has an inverse
that is also a function!!!
This inverse is called the
logarithmic function with base b.
Notation:

3. a logarithm is an exponent!!!
Changing Between Logarithmic
and Exponential Form
If x > 0 and 0 < b = 1, then
if and only if
Important Note: The “linking statement” says that
a logarithm is an exponent!!!

4. Basic Properties of Logarithms
For 0 < b = 1, x > 0, and any real number y, 1.
because 2.
because 3.
because 4.
because

5. Evaluating Logarithms
Evaluate each of the following. 1. 4. 2. 5. 3. 6.

6. Now, let’s plot these points and discuss the graphs…
What’s true about the (x, y) pairs and graphs of inverse functions? –3
1/8
1/8 –3 –2
1/4
1/4 –2 –1
1/2
1/2 –1 1 1 1 2 2 1 2 4 4 2 3 8 8 3
Now, let’s plot these points and discuss the graphs…

7. Common Logarithms Common Logarithm – logarithm with a base of 10
(very commonly used because of our base 10 number system!)
For common logarithms, we can drop the subscript:
if and only if

8. Basic Properties of Common Logarithms
Let x and y be real numbers with x > 0. 1.
because 2.
because 3.
because 4.
because

9. More Evaluating Logarithms
Evaluate each of the following. 1. 2. 3. 4.
Note: The LOG key on your calculator refers
to the common logarithm…

10. Using Your Calculator b/c b/c is undefined  can you explain why ?
Use a calculator to evaluate the logarithmic expression if it is
defined, and check your result by evaluating the corresponding
exponential expression. 1.
b/c 2.
b/c 3.
is undefined  can you explain why ?

11. Solving Simple Logarithmic Equations
Solve the given equations by changing to exponential form. 1. 2.
Exp. Form:
Exp. Form:

12. What is the definition of the natural base???

13. Natural Logarithms Natural Logarithm – a logarithm with base e
Notation: ln
That is,
Back to our inverse relationship:
if and only if

14. Basic Properties of Natural Logarithms
Let x and y be real numbers with x > 0. 1.
because 2.
because 3.
because 4.
because

15. Cool Practice Problems
Evaluate each of the following without a calculator. 1. 3. 2.
Note: The LN key on your calculator refers
to the natural logarithm…

16. Cool Practice Problems
Use a calculator to evaluate the given logarithmic expressions,
if they are defined, and check your result by evaluating the
corresponding exponential expression. 1.
because 2.
is undefined!!! Why??? 3.
because

17. Cool Practice Problems
Solve each of the given equations by changing them to
exponential form. 1. 2. 1
x = 100,000
x = = 0.368… e 3. 4. 1
2.5
x = = 0.01
x = e = …
100

18. Analysis of the Natural Logarithmic Function:
The graph:
Domain:
Range:
Continuous on
Increasing on
No Symmetry
Unbounded
No Local Extrema
No Horizontal Asymptotes
Vertical Asymptote:
End Behavior:

19. Note: Any other logarithmic function
Analysis of the Natural Logarithmic Function
The graph:
Note: Any other logarithmic
function
with b > 1 has the same domain,
range, continuity, inc. behavior,
lack of symmetry, and other
general behavior of the natural
logarithmic function!!!

20. Reflect across the y-axis, Trans. right 3  The graph?
Describe how to transform the graph of y = ln(x) or y = log(x)
into the graph of the given function. Sketch the graph by hand
and support your answer with a grapher. 1.
Trans. left 2  The graph? 2.
Reflect across the y-axis,
Trans. right 3  The graph?

21. Vert. stretch by 3  The graph?
Describe how to transform the graph of y = ln(x) or y = log(x)
into the graph of the given function. Sketch the graph by hand
and support your answer with a grapher. 3.
Vert. stretch by 3  The graph? 4.
Trans. up 1  The graph?

22. Trans. right 1, Horizon. shrink by 1/2,
Describe how to transform the graph of y = ln(x) or y = log(x)
into the graph of the given function. Sketch the graph by hand
and support your answer with a grapher. 5.
Trans. right 1, Horizon. shrink by 1/2,
Reflect across both axes, Vert. stretch by 2,
Trans. up 3  The graph???

23. 1. Graph the given function, then analyze it for domain, range,
continuity, increasing or decreasing behavior, boundedness,
extrema, symmetry, asymptotes, and end behavior. 1. D: R:
Continuous
Dec:
No Symmetry
Unbounded
No Local Extrema
Asy:
E.B.:

24. 2. Graph the given function, then analyze it for domain, range,
continuity, increasing or decreasing behavior, boundedness,
extrema, symmetry, asymptotes, and end behavior. 2. D: R:
Continuous
Dec:
No Symmetry
Unbounded
No Local Extrema
Asy:
E.B.: