Name: ________________________________
Ch. 11 WS #4
Date: ______________
Hr: ______
Undoing Exponents: Logarithms
Inverse Functions: Recall that an inverse function UNDOES
the effects of the original function.
1. What kind of function would undo a power function?
(a) Suppose y = x2. What is the opposite of squaring x? (That is,
what is its inverse function?)
(b) Sketch a graph of both y = x2 and its inverse function on the
same set of axes.
(c) Suppose y = x3. What is the opposite of cubing x? (That is,
what is its inverse function?)
(d) Sketch a graph of both y = x3 and its inverse function on the
same set of axes.
(e) How is the graph of a function related to a graph of its inverse function?
2. What kind of function would undo an exponential function such as y = 2x? Think about this
by completing the following:
(a) Fill in the table below using the function y = 2x:
x
-3
-2
-1
0
1
2
y
1/8
3
(b) The inverse function of y = 2x will take all the input and output values
and switch them. Fill in the table of values for the INVERSE
FUNCTION of y = 2x below. [Hint: Just switch the x’s and y’s from
the table above].
x
1/8
y
-3
(c) Sketch a graph of both y = 2x and its inverse function on the same set of axes.
Logarithms: The function that UNDOES an exponential function is called a “logarithm”.
The “base” of the logarithm corresponds to the base of the exponential function which
it is undoing. By definition, then…
If y = bx
then
logby = x
…where here log stands for “logarithm” and b is its base.
When trying to find the value of logarithms, you must basically think to yourself, “b to
WHAT POWER will make y??”
Example: Suppose you must find log216. This is asking, “2 to WHAT POWER will make
16?” The answer would be 4, since 24 = 16. Therefore, log216 = 4.
3. Carefully read the example above to evaluate the following. (Careful: Although there is a
“log” key on your calculator, it is only for log base 10. You will have to THINK to do these.)
(a)
log416 = _____
(k)
log44 = _____
(b)
log327 = _____
(l)
log5125 = _____
(c)
log264 = _____
(m)
log71 = _____
(d)
log4(1/4)= _____
(n)
log3243= _____
(e)
log6(1/36)= _____
(o)
log101000= _____
(f)
log28 = _____
(p)
log864 = _____
(g)
log7√ = _____
(q)
log63√
(h)
log2(1/8)= _____
(r)
log100.1 = _____
(i)
log4√
(s)
log100.01 = _____
(j)
log42 = _____
(t)
log20.25 = _____
-3
1
-2
3/2
= _____
Answers to #3: (May be used more than once!!!)
= _____
-1
2
0
3
1/3
5
½
6
4. To help answer some of the following questions, it will help to review some common
powers. Fill in the following:
(a) 2-3 = _____
2-2= _____
2-1= _____
20= _____
21= _____
23= _____
24 = _____
25= _____
26= _____
27= _____
28= _____
29= _____
(b) 3-3 = _____
3-2= _____
3-1= _____
30= _____
31= _____
33= _____
34 = _____
35= _____
36= _____
37= _____
38= _____
39= _____
4. (cont’d)
(c) 4-3 = _____
4-2= _____
4-1= _____
40= _____
41= _____
44 = _____
45= _____
46= _____
47= _____
48= _____
(d) 5-3 = _____
5-2= _____
5-1= _____
50= _____
51= _____
54 = _____
55= _____
56= _____
57= _____
43= _____
53= _____
5. Finding A Pattern: Logs of Multiplied Numbers Evaluate the following and look for a
pattern. SHOW YOUR INTERMEDIARY STEPS AS IN THE EXAMPLE!!
(a) log2(8·4) = _log2(32) = 5
log28 + log24 = _3 + 2 =__________
(b) log2(32·2) = _________________
log232 + log22 = __________________
(c) log3(27·9) = _________________
log327 + log39 = __________________
(d) log3[81·(1/9)] = _________________
log381 + log3(1/9) = ________________
(e) log4(64·16) = _________________
log464 + log416 = __________________
(f) log5(25·125) = _________________
log525 + log5125 = _________________
(g) What pattern do you notice?
logb(n·m) = ___________________
6. Finding A Pattern: Logs of Divided Numbers Evaluate the following and look for a pattern.
SHOW YOUR INTERMEDIARY STEPS!!
(a) log2(16/2) = _log2(8) = 3
log216 – log22 = _4 – 1 =__________
(b) log2(128/32) = _________________
log2128 – log232 = _________________
(c) log3(729/81) = _________________
log3729 – log381 = _________________
(d) log3(9/81) = _________________
log39 – log381 = ________________
(e) log4(16384/4096) = _______________
log416384 – log44096 = _____________
(f) log5(625/125) = _________________
log5625 – log5125 = ________________
(g) What pattern do you notice?
logb(n/m) = ___________________
7. Finding A Pattern: Logs of Numbers to a Power Evaluate the following and look for a
pattern. SHOW YOUR INTERMEDIARY STEPS!!
(a) log2(45) = _log2(1024) = 10
5·log2 4 = _5·2 =_10_________
(b) log2(43) = _________________
3·log2 4= _________________
(c) log3(94) = _________________
4·log39 = _________________
(d) log3(272) = _________________
2·log327 = ________________
(e) log4(210) = _______________
10·log42 = _____________
(f) log5(252) = _________________
2·log525 = ________________
(g) What pattern do you notice?
logb(np) = ___________________
8. A Few More Patterns: Think about it….
(a) What must ALWAYS be any log base of the number 1?
logb1 = _________
(b) What must ALWAYS be any log base b of the number b?
logbb = _________
9. Use the properties you discovered in questions #5-7 to simplify the following as much as
possible. Assume all logs are of the same base, so the base has not been listed. Note that
logs can be considered as “like terms”, and like terms can be combined. For example
2 log A + log A = 3 log A
(a) log A3 – log B2/3 + log A1/3 + log B5/3
(b) log(ABC)
log( )
(c)
log A2 – 2 log B
log A2 + log ( )