Chapter 4
Scientific Notation, Exponents, and Logarithms
4.1
Scientific Notation
Scientific notation expresses large numbers and
small numbers using powers of ten: Therefore,
3250000000 = 3.25 x 10 9 and 0.00000145 =
1.45 x 10-6. This can be done on the calculator
using either 10x , the power of ten key or EE ,
Figure 4.1
the exponent of ten key.
Note: Any decimal number can be written in
scientific notation using the form, where
1  K  10 and x is an integer. To change back
to decimal number form: For K x 10x , if x >0
move x decimal places to the right, if x < 0 move
x decimal places to the left.
Figure 4.2
Example 1
Type the following numbers:
a. 3.25  10 9
b. 0.00000145
Press 3.25 2nd
10x 9 )
press 3.25 2nd
EE 9 ENTER . Compare
ENTER , then
Figure 4.3
the results. See Figure 4.1. When .00000145 is
typed, it is changed automatically to scientific
notation.
Troubleshooting:
For 10x when x  10 or x  4 , the number
is written in scientific notation. See Figures 4.2
and 4.3.
4.2
Figure 4.4
Verifying Properties of
Exponents
Example 2
Verify numerically that the following are true
by choosing values for a.
a. a 0  1
1
b. a 1 
a
c.
a6  a a a a a a
Let a assume various values. Enter the
problems as in Figures 4.4 and 4.5. Press ^ for
the exponent key.
Figure 4.5
4-2
Explorations In College Algebra 5e: Graphing Calculator Manual
Chapter 4
4.2.1
Other Exponent Keys
There are other shortcut keys for exponents. The
x2 and x -1 keys are used to paste the
exponents without using the ^ key . See
Figure 4.6. The cubic power and radical symbol
Figure 4.6
are found under MATH . See Figure 4.7.
4.2.2
Fractional Exponents
Fractional exponents can be written as radicals.
For any number a, then a 1 n  n a . (Note: n is
called the root index.)
Figure 4.7
Example 3
Show that the following are equivalent.
a.
251 / 2  25
b.
321 / 5  5 32
Troubleshooting: Fractional exponents must
be enclosed in parentheses.
For 251 / 2 press 25 ^
ENTER . For
( 1 / 2)
25 , press 2nd
25 )
Figure 4.8
ENTER . See Figure 4.8. Roots other than
square root are found under the MATH menu.
You must always type the root index first. See
5
Figure 4.7. To type
select [ 5 : x
(32 ) , press 5 MATH ;
] ; type ( 32 )
ENTER .
Figure 4.9
See Figure 4.9.
Note: For fractional exponents
am/n  n am 
4.3
 a
n
m
. See Figure 4.10.
Using the Logarithm Key
To find the exponent or power of ten in an
equation, we use logarithms to “undo” the
exponent.
If 10 x  N then log 10 N  x
Example 4
Write 10x = 25 as a logarithmic equation.
A logarithm is the value of the exponent or
power of the base, which in this case is 10. The
logarithmic equation is: log 10 25 = x . You read
the above equation as “The logarithm of 25 base
10 is x” or “log 25 is x.”
Figure 4.10
Chapter 4
To find the value of the exponent press LOG
25 ) . See Figure 4.11.
Note: log 25 = log 10 25. This is the “common
logarithm”. The base 10 is understood and
conventionally not written.
Figure 4.11
To check your work:
101.397940009 ˜ 25
Press 10 ^
2nd
ANS
ENTER .
See Figure 4.12.
Figure 4.12
Note: If you type in a rounded off value of the
log 25 you will get close to 25 but not exactly
25. A logarithm is a decimal number that is nonterminating and non-repeating. See Figure 4.12.
Figure 4.13
Example 5
Sketch the graph of y= log x and use it to
determine the domain and range of the
function.
Press Y=
CLEAR
LOG
X , T , , n
Figure 4.14
) . See Figure 4.13.
Graph the function. Press ZOOM ; select
[4:Zdecimal]; press TRACE . The graph in
Figure 4.14 shows that y  log x is undefined
when x = 0. Use

Figure 4.15
to confirm that
y  log x is undefined for x < 0 . See Figure
4.15. The domain of y  log x is the set of all x
such that x > 0 or in interval notation x : (0, ) .
The range is evident by looking at the graph,
also. We see that as x increases y increases, but
what happens as x approaches zero? y seems to
be headed in a negative direction. See
calculations in Figure 4.16. The range for
y  log x is the set of all real numbers, or in
interval notation y : (, ) .
Figure 4.16
Figure 4.17
Verify the range values by using 2nd
TblSet . Start at 0 and increment by .0001.
See Figure 4.17. Use 2nd
TABLE to see the
values for y  log x . We see that for 0 < x < 1,
y is getting more negative slowly . See Figure
4.18.
Figure 4.18
4-3
4-4
Explorations In College Algebra 5e: Graphing Calculator Manual
4.4
Solving Equations Graphically
Chapter 4
We saw in Chapter 3 that the point of
intersection represents the solution to an
equation. You can solve an equation graphically
by locating the point of intersection.
Figure 4.19
Example 6
Solve the equation 10x = 25 graphically.
Enter the following into Y= :
Y1 = 10x
Y2 = 25
See Figure 4.19.
Figure 4.20
Press WINDOW . Set the WINDOW so
that both equations can be seen. See Figure
4.20.
Press GRAPH . See Figure 4.21.
The point of intersection represents the
solution to the equation. Press 2nd
CALC ;
Figure 4.21
select [5:intersect] . See Figure 4.22. Follow
the prompts by pressing ENTER . The point
of intersection occurs at approximately x
=1.39794. See Figure 4.23.
Note: The calculator remembers the intersection
value for x. Immediately go to the Home
Screen. Press 2nd
Figure 4.22
QUIT X , T , , n . See
Figure 4.24.
Verify the solution by typing the expression
10 ^ X , T , , n . See Figure 4.24.
So we see that when x  1.39794 then
1.39794
10
 25 . This makes sense since
1
10  10 and 10 2  100 so 10 x  25 is
somewhere in between, so the power of ten, x,
must be 1  x  2 .
Figure 4.23
Figure 4.24