Alg II Notes – More about Logs and Graphs of Inverse Functions (including logs)
Since the numerical system used everyday is base 10 (ones, tens, hundreds, etc), base 10
logarithms are particularly important. Base 10 logarithms are called common logarithms. Use a
shorthand notation of omitting the base when indicating base 10 logarithms. So, when you see
“log 100,” it means log10 100 . (Note: you calculator only has buttons for 2 bases of logs; one is base 10,
the other is base e, which we will learn about later).
What is the value of each of the following:
ex 1 - log 10
____________ (This means log10 10 Or “10 to what power is 10?”)
ex 2 - log 10000 ____________ (This means log10 10000 Or “10 to what power is 10000?”)
ex 3 - log 0.001 ____________ (This means log10 .001 Or “10 to what power is .001?”)
Now do some thinking in each of the following sets of examples. Write the actual value of the first
two numbers and then estimate the third. (Check your estimate using your GC; were you close?)
ex 4 - log 100
__________
ex 5 - log 1
__________
log 1000 __________
log 10 __________
log 300
log 5 __________
__________
1. Common logarithms are base ____ logarithms.
2. Simplify: log100
log .01
log 1017
3. Which of the following are one-to-one functions (there may be more than 1)? _____________
x7
y
A.) y  x  3
B.)
3
y
y
4
4
2
2
–4
C.)
–2
2
4
–4
x
–2
2
–2
–2
–4
–4
D.)
4
x
Graphing Inverses To graph an inverse, find points from the original graph then switch the x and y
coordinates. The inverse graph should be reflected over the y = x line. The Domain and the Range will
also be switched.
Draw the inverse of each given graph on the same plane. Then identify the Domain/Range for the original
and its inverse.
y
–4
y
y
4
4
4
2
2
2
–2
2
x
4
–4
–2
2
4
x
–4
–2
2
–2
–2
–2
–4
–4
–4
4
x
Domain of f(x)__________
Range of f(x) ___________
Domain of f(x)__________
Range of f(x) ___________
Domain of f(x)__________
Range of f(x) ___________
Domain of f-1 (x) _________
Range of f-1 (x) _________
Domain of f-1 (x) _________
Range of f-1 (x) __________
Domain of f-1 (x) _________
Range of f-1 (x) _________
5. Recall your work with Inverse variation. We learned that the graph for inverse variation looked like a
hyperbola. Graph the inverse variation xy  12 on the grid below. You may use the table at the right to help.
x
y
_________________
What do you notice about this graph and its inverse?
What is the domain (for each)?
What is the range (for each)?
Give the equation for the vertical asymptote:
For the horizontal asymptote:
True or False: Sometimes a function is its own inverse _______
y
12
10
8
6
4
2
–12
–10
–8
–6
–4
–2
2
–2
–4
–6
–8
–10
–12
4
6
8
10
12
x
Algebra 2
A.26
Name __________________________
Homework-Graphing inverses & converting log/expon
Draw the inverse of each given graph on the same plane. Then identify the Domain/Range for the original
and its inverse.
1.
2.
3.
y
–4
y
y
4
4
4
2
2
2
–2
2
x
4
–4
–2
2
4
x
–4
–2
2
4
–2
–2
–2
–4
–4
–4
Domain of f(x)__________
Range of f(x) ___________
Domain of f(x)__________
Range of f(x) ___________
Domain of f(x)__________
Range of f(x) ___________
Domain of f-1 (x) _________
Range of f-1 (x) _________
Domain of f-1 (x) _________
Range of f-1 (x) __________
Domain of f-1 (x) _________
Range of f-1 (x) _________
x
4. The graph for y  log3 x is shown below. Graph its inverse.
y
5. What is the domain for y  log3 x ? The range?
8
6
4
6. What is its inverse?
2
–8
–6
–4
–2
2
4
6
8
x
–2
–4
7. What is the domain of the inverse?
–6
–8
What is the range of the inverse?
Is it a function?
8. Graph y  2 x  3 Now graph its inverse. Which of the following points would not lie on the graph of the
inverse?
(-5, 4) (-1, 2) (0, -3) (5, -1)
y
9
x
-9
9
-9
9. Graph y  x 3
inverse?
Now graph its inverse. Which of the following points would not lie on the graph of the
(-8, -2) (-1, -1) (0, 0) (8, 2)
y
9
x
-9
9
-9
10. What is the inverse of f ( x)  3x ?
11. What is the inverse of f ( x)  9 x 2 ?
12. If the asymptote for a function, f , is a horizontal line y  3 , the asymptote for its inverse will be a
_________________ line and its equation will be ____________ .