2. Real life situations of the
exponents
The Internet is growing faster
than all other technologies
that have preceded it.
Radio existed for 38 years
before it had 50 million
listeners.
Television took 13 years to
reach that mark.
The Internet crossed the line in
just four years.
3. Objective
At the conclusion of this slide show you will be able
to the following:
Write numbers in exponential form
Evaluate expressions with exponents
Evaluate expressions with negative exponents
And evaluate the zero exponent
4. Vocabulary
Base – the number that is being used as a factor.
Exponent – tells you how many times the base is to be multiplied.
Exponential Form – when a number is written with a base and an
exponent.
Power – the number that is produced by raising a base to an
exponent.
Reciprocal – is 1 divided by the number itself
5. Writing exponents
How do I write something in exponential form?
4•4•4•4•4•4•4•4
Identify how many times 4 is being used as a factor.
(In this case 4 is being used as a factor 8 times, so 8 would be
our exponent.)
Write in exponential form, 4 • 4 • 4 • 4 • 4 • 4 • 4 • 4
Would be written as
6. Now you try
1.
2.
3.
4.
Write the following in exponential form
3•3•3•3
x•x•x•x•x•x
7•7•7•y•y•y•y
(- 4) • (- 4) • (- 4) • (- 4) • (- 4) • (- 4)
8. How do I evaluate powers?
Simply by multiplying your base by the number of
times indicated by the exponent.
So you would find the product of 5 being multiplied
three times. 5 • 5 • 5 = 125
Watch negative numbers
means you are
multiplying -5 twice. So (-5) • (-5) = 25
And
means you are multiplying 5 twice and
making your answer negative
– (5 • 5 ) = -25
11. What’s up with negative exponents?
Any negative exponent can be written as the
reciprocal of the base raised to the positive
exponent.
What that means in plain English is that
can be written as
, now to evaluate this
you would multiply • . =
14. Can zero be an exponent(important
question)?
YES it can! He is how zero works as an exponent.
Now this is very complicated…
and you need to commit this to memory…
19. Exponential Functions
A function is called an exponential function if it
has a constant growth factor.
This means that for a fixed change in x, y gets
multiplied by a fixed amount.
Example: Money accumulating in a bank at a
fixed rate of interest increases exponentially.
20. One-to-One Functions
A function is a one-to-one function if each value in the range corresponds
with exactly one value in the domain.
For a function to be one-to-one, it must not only pass the
vertical line test, but also the horizontal line test.
y
y
x
Function
x
Not a one-to-one
function
y
x
One-to one
function
22. Other Examples of
Exponential Functions
Populations tend to growth exponentially not linearly.
When an object cools (e.g., a pot of soup on the
dinner table), the temperature decreases
exponentially toward the ambient temperature.
Radioactive substances decay exponentially.
Viruses and even rumors tend to spread exponentially
through a population (at first).
23. Reflected about y-axis
y2
x
This equation could be rewritten in a different
form:
y2
x
1 1
x
2
2
x
So if the base of our exponential function is
between 0 and 1 (which will be a fraction),
the graph will be decreasing. It will have the
same domain, range, intercepts, and
asymptote.
These two exponential functions have special names.
24. Let’s examine exponential functions. They are different than any
of the other types of functions we’ve studied because the independent
variable is in the exponent.
Let’s look at the graph of this
function by plotting some points.
x
3
2
1
0
-1
1/2
-3
1/8
2x
8
4
2
1
f x 2
x
BASE
Recall what a negative
exponent means:
f 1 2 1
1
2
8
7
6
5
4
3
2
1
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
-2
-3
-4
-5
-6
-7
25. Inverse Functions
If f(x) is a one-to-one function with ordered pairs of the form (x,y), its inverse
function, f -1(x), is a one-to-one function with ordered pairs of the form (y,x).
Function:
{(2, 6), (5,4), (0, 12), (4, 1)}
Inverse Function:
{(6, 2), (4,5), (12, 0), (1, 4)}
• Only one-to-one functions have inverse functions.
• Note that the domain of the function becomes the
range of the inverse function, and the range
becomes the domain of the inverse function.
26. u
a
If
=
v,
a
then u = v
This says that if we have exponential functions in equations and we can
write both sides of the equation using the same base, we know the
exponents are equal.
2
3 x4
3 x 4
2
8
2
3x 4 3
3
The left hand side is 2 to the something. Can we rewrite the right hand side as 2 to the something?
Now we use the property above. The bases are both 2
so the exponents must be equal.
We did not cancel the 2’s, We just used the property
and equated the exponents.
You could solve this for x now.
27. Guidelines to solve inverse functions
To Find the Inverse Function of a One-to-One Function
1. Replace f(x) with y.
2. Interchange the two variables x and y.
3. Solve the equation for y.
4. Replace y with f –1(x). (This gives the inverse function
using inverse function notation.)
Example:
Find the inverse function of f x x 1, x 1.
Graph f(x) and f(x) –1 on the same axes.
29. The logarithmic function to the base a, where a > 0 and a 1
is defined:
y = logax if and only if x = a y
logarithmic
form
exponential
form
When you convert an exponential to log form, notice that the
exponent in the exponential becomes what the log is equal to.
Convert to log form:
16 4
2
log 416 2
Convert to exponential form:
1
log 2 3
8
2
3
1
8
30. How to convert logarithms to exponents
log 2 16 4
This is asking for an exponent. What
exponent do you put on the base of 2 to
get 16? (2 to the what is 16?)
1
log 3 2
9
What exponent do you put on the base of
3 to get 1/9? (hint: think negative)
log 4 1 0
1
2
1
log
log 33 33
2
What exponent do you put on the base of
4 to get 1?
When working with logs, re-write any
radicals as rational exponents.
What exponent do you put on the base of
3 to get 3 to the 1/2? (hint: think rational)
31.
32. Solve for x: log 6 x 2
Solution:
Let’s rewrite the problem in
exponential form.
6 x
2
We’re finished !
33. 1
Solve for y: log 5
y
25
1
5
25
5y 5 2
y
y 2
1
Since 5 2
25
35. Objective Check
You should now be able to explain the following terms in your own words :
Base
Exponent
Exponential Form
Power
Reciprocal
Draw exponential function
You should also be able to:
Write numbers in exponential form
Evaluate expressions with exponents
Evaluate expressions with negative exponents
And evaluate the zero exponent
Distinguish and solve inverse functions
