1. LOGARITHMS Section 4.2
JMerrill, 2005
Revised 2008

2. Exponential Functions
1. Graph the exponential equation f(x) = 2x on the graph and record some ordered pairs. x
f(x) 1 2 4 3 8

3. Review 2. Is this a function? 3. Domain?
Yes, it passes the vertical line test (which means that no x’s are repeated)
3. Domain?
Range?

4. Review
2. Is the function one-to-one? Does it have an inverse that is a function?
Yes, it passes the horizontal line test.

5. Inverses
To graph an inverse, simply switch the x’s and y’s (remember???)
f(x) = f -1(x) = x
f(x) 1 2 4 3 8 x
f(x) 1 2 4 8 3

6. Now graph
f(x)
f-1(x)

7. How are the Domain and Range of f(x) and f -1(x) related?
The domain of the original function is the same as the range of the new function and vice versa.
f(x) = f -1(x) = x
f(x) 1 2 4 3 8 x
f(x) 1 2 4 8 3

8. Graphing Both on the Same Graph
Can you tell that the
functions are inverses
of each other? How?

9. Graphing Both on the Same Graph
Can you tell that the
functions are inverses
of each other? How?
They are symmetric
about the line y = x!

10. Logarithms and Exponentials
The inverse function of the exponential function with base b is called the logarithmic function with base b.

11. Definition of the Logarithmic Function
For x > 0, and b > 0, b  1
y = logbx iff by = x
The equation y = logbx and by = x are different ways of expressing the same thing. The first equation is the logarithmic form; the second is the exponential form.

12. Location of Base and Exponent
Logarithmic: logbx = y
Exponential: by = x
Exponent
Exponent
Base
Base
The 1st to the last = the middle

13. Changing from Logarithmic to Exponential Form
a. log5 x = 2 means 52 = x
So, x = 25
b. logb64 = 3 means b3 = 64
So, b = 4 since 43 = 64
You do:
c. log216 = x means
So, x = 4 since 24 = 16
d. log255 = x means
So, x = ½ since the square root of 25 = 5!
2x = 16
25x = 5

14. Changing from Exponential to Logarithmic
a. 122 = x means log12x = 2
b. b3 = 9 means logb9 = 3
You do:
c. c4 = 16 means
d. 72 = x means
logc16 = 4
log7x = 2

15. Properties of Logarithms
Basic Logarithmic Properties Involving One:
logbb = 1 because b1 = b.
logb1 = 0 because b0 = 1
Inverse Properties of Logarithms:
logbbx = x because bx = bx
blogbx = x because b raised to the log of some number x (with the same base) equals that number

16. Characteristics of Graphs
The x-intercept is (1,0). There is no y-intercept.
The y-axis is a vertical asymptote; x = 0.
Given logb(x), If b > 1, the function is increasing. If 0<b<1, the function is decreasing.
The graph is smooth and continuous. There are no sharp corners or gaps.

17. Transformations Vertical Shift
Vertical shifts
Moves the same as all other functions!
Added or subtracted from the whole function at the end (or beginning)

18. Transformations Horizontal Shift
Horizontal shifts
Moves the same as all other functions!
Must be “hooked on” to the x value!

19. Transformations Reflections
g(x)= - logbx
Reflects about the x-axis
g(x) = logb(-x)
Reflects about the y-axis

20. Transformations Vertical Stretching and Shrinking
f(x)=c logbx
Stretches the graph if the c > 1
Shrinks the graph if
0 < c < 1

21. Transformations Horizontal Stretching and Shrinking
f(x)=logb(cx)
Shrinks the graph if the c > 1
Stretches the graph if
0 < c < 1

22. Domain
Because a logarithmic function reverses the domain and range of the exponential function, the domain of a logarithmic function is the set of all positive real numbers unless a horizontal shift is involved.

23. Domain Con’t.
Domain
Domain
Domain

24. Properties of Commons Logs
General Properties
Common Logarithms
(base 10)
logb1 = 0
log 1 = 0
logbb = 1
log 10 = 1
logbbx = x
log 10x = x
blogbx = x
10logx = x

25. Properties of Natural Logarithms
General Properties
Natural Logarithms
(base e)
logb1 = 0
ln 1 = 0
logbb = 1
ln e = 1
logbbx = x
ln ex = x
blogbx = x
elnx = x