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In this lesson, exponential functions with base and the key features of their graphs will be discussed. Furthermore, logarithmic functions will be introduced and graphed. Finally, the relationship between exponential and logarithmic functions will be explained both algebraically and graphically.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.


Explore

Identifying Change

As soon as Zosia entered the room for her Algebra lesson, her teacher asked a question to the class.
tables
Help Zosia find the answer by considering the following questions.
Discussion

Constant Multiplier

The rate of change for linear functions is constant. For each step in the direction, the change between values is the same. Therefore, the difference between values for consecutive values is the same. Conversely, the rate of change for exponential functions is not constant. This means that the differences between values for consecutive values are not the same.

In the table on the left, the difference between values for consecutive values is This means that the rate of change is constant. Conversely, in the table on the right, the differences between values for consecutive values are not the same. This means that for this table the rate of change is not constant.

The table on the left represents a linear function because it shows a constant rate of change. Conversely, the table on the right represents a non-linear function. Notice how the values, for the table on the right, are doubled for each step in the direction. To obtain a value, the previous value is multiplied by

The table on the left corresponds to a function with a constant increase. In the table on the right, the increase is not constant. However, the ratio between values for consecutive values is always the same. This means that the table on the right corresponds to a function with a constant multiplier. Functions with constant multipliers are exponential functions.
Pop Quiz

Identifying the Constant Multiplier

Does the table below correspond to an exponential function? If yes, write only the constant multiplier. If not, write only no.

Is it an exponential function?
Discussion

Natural Base Exponential Functions

Natural base exponential functions are exponential functions whose base, or constant factor, is the number .
Here, and are real numbers that are not equal to In considering their positive or negative sign, two observations can be made.

The graphs of and are shown.

In general, the graph of a natural base exponential function is always a smooth curve.
Method

Graphing Natural Base Functions

A natural base exponential function with both and equal to will be considered rather than more complex values for the sake of understanding the basics.
To graph this function, three steps must be followed.
1
Identify the Initial Value and Plot the
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The initial value of an exponential function, is the number without an exponent, or the value of The initial value is also known as the intercept In this example, the value of is and thus the intercept of the graph of is

y-intercept of y=e^x
2
Use the Constant Multiplier to Find More Points
expand_more

In natural base functions, the constant multiplier — the number with an exponent — is the number This means that when values increase by the values are multiplied by With this information, more points can be plotted on the coordinate plane. It is advised to have not less than three points.

plotting points
3
Draw the Curve
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Lastly, the graph can be drawn by connecting the points with a smooth curve.

Graph of the function y =e^x with points plotted at (0,1), (1,2.7), and (2,7.4)

It is worth remembering that, in general, the graph of a natural base function is always a smooth curve.

Rule

Continuously Compounded Interest

When interest is compounded infinitely many times, it is said to be continuously compounded. Let be the balance of an account that is continuously compounded, the initial amount, the interest rate, and the time. These values are connected by the following formula.

Keep in mind that, in this formula, the value of must be written as a decimal and the time must be in years. Also, the initial amount is usually called principal.

Proof
Recall the compound interest formula.
In the formula, there is the account's balance , the initial amount the time in years and the number of times the interest is compounded per year. The formula can be rewritten by using the Power of a Power Property.
If the interest rate is or and the interest is continuously compounded, the formula can be written in terms of and This is because as goes to infinity, the value of approaches
The value of the expression , on the other hand, approaches Examine the following interactive graph to better grasp how this is possible.
graphs of $y =e^r$ and $y=(1+r/n)^n$
Using this last approximation, the final form of the formula can be obtained.
Example

Graphing and Using a Continuously Compounded Interest Graph

Zosia wants to visit family in Argentina, and while there she hopes to climb Mount Aconcagua! This is going to cost a pretty penny so she needs to increase her savings. Zosia knows that banks do not offer continuously compounded interest, but she is daydreaming about opening an account that does use it.

ATM
Recently, she made from selling some old stuff at a garage sale. She would deposit this amount into the account. The amount in the account after years is calculated by using the following formula.
a Graph the given function.
b Use the formula to calculate the balance of this account after year and months. Suppose that no deposits nor withdrawals are made and round the answer to two decimal places. Check the answer with the graph.
c Use the graph to estimate in how many years the balance of this account would be Suppose that no deposits nor withdrawals are made. Round the answer to the nearest integer.

Answer
a
answer
b About
c About years

Hint
a Start by identifying the initial value and plotting the intercept.
b Keep in mind that year and months are years.
c Identify the point on the curve whose second coordinate is

Solution
a The given function is a natural base exponential function whose initial value is
Therefore, the intercept of the graph occurs at Since time cannot be negative and the balance of the account will always be non-negative, only the first quadrant of the coordinate plane will be considered.
y-intercept

The constant multiplier is This means that when values increase by the values are multiplied by

y-intercept,(1,46.1), and (2,125.6)

Finally, the curve can be graphed by connecting these points.

Graph of A=17e^t
b First, year and months must be expressed in years. Recall that year has months. This means that months are or years.
Therefore, will be substituted into the equation.
Substitute

UseCalc

Use a calculator

RoundDec

Round to decimal place(s)

After year and months, the balance of the account would be about This can be verified by using the graph. To do so, the point whose first coordinate is should be determined on the graph of the function.
spotting the point with the first coordinate 1.75

The second coordinate of this point is a bit less than Therefore, it is reasonable to have a balance of after year and months.

c This time the graph will be used to estimate after how many years the balance would be To do so, the point with the second coordinate must be spotted on the graph of the function, and its first coordinate needs to be identified.
spotting the point with the second coordinate 120

The first coordinate of the point is almost equal to Therefore, the balance would be after about years. Zosia is realizing that it might take her quite some time to meet her financial goal, but she is making a valiant effort and certainly on her way!

Example

Using a Natural Base Exponential Function to Model the Atmospheric Pressure at Different Altitudes

Zosia finally saved enough money to visit family in Argentina and set out to climb Mount Aconcagua while there! She is as excited as ever.

Error loading image.
Here, her knowledge about natural base exponential functions will be extremely useful to her pursuit of reaching the peak. Zosia knows that the atmospheric pressure decreases as the altitude increases, the pressure at sea level is about hecto Pascals (hPa), and the pressure at an altitude of meters is given by the following formula.
To avoid unforeseen problems, Zosia realizes that she can share some responsibilities if she recruits a team to solve such issues. Give her a hand and become a valuable member of her team by answering the following questions. If unable to, the team could get altitude sickness!
a The altitude of the mountain's peak, called the summit, is meters. Calculate the atmospheric pressure at the summit. Round the answer to the nearest integer.
b Draw a graph of the given exponential function.
c Estimate the altitude where the atmospheric pressure is hecto Pascals. At this altitude, Zosia will rest for a night or two. Round the answer to the nearest thousand meters.

Answer
a About hPa
b
graph of the function
c About m

Hint
a Substitute for in the given formula.
b Make a table of values.
c Use the graph from Part B.

Solution
a To calculate the atmospheric pressure at an altitude of meters, this number will be substituted for into the given function.
Substitute

▼
Evaluate right-hand side
MultNegPos

UseCalc

Use a calculator

RoundInt

Round to nearest integer

The pressure at the summit of Mount Aconcagua, which has an altitude of meters, is about hecto Pascals.
b To graph the given function, a table of values will be made first. Only non-negative values will be considered since it is a mountain climb under consideration.

The obtained points will now be plotted on a coordinate plane and connected with a smooth curve. Since only non-negative values are considered, the graph just needs to be drawn in the first quadrant.

graph of the function
c To estimate the altitude where the atmospheric pressure is hecto Pascals, the graph from Part B will be used. The point with the second coordinate will be spotted and its coordinate identified.
graph of the function

Tracing a pen or pencil vertically down from the point of the graph that is even, horizontally, with of leads to the of This means that the altitude with an atmospheric pressure of hecto Pascals is about meters. Zosia feels comfortable knowing she will be able to rest for a few nights at this level.

Discussion

Logarithmic Functions

Logarithmic functions are functions that involve logarithms.

The function is the parent function of logarithmic functions. Since logarithms are defined for positive numbers, the domain of the function is and its range is all real numbers. If is less than the graph of the function is decreasing over its entire domain. Conversely, if is greater than the graph is increasing over its entire domain.

Rule

Inverse Properties of Logarithms

A logarithm and a power with the same base undo each other.

In particular, the above equations also hold true for common and natural logarithms.

Proof

The general equations will be proved one at a time.

This identity can be proved by using the Power Property of Logarithms and the definition of a logarithm.

IdPropMult

Identity Property of Multiplication

\(x\ {\color{#009600}{\bm{\Large{\checkmark }}}}\)
The logarithm of with base is equal to

Let Therefore, by the definition of a logarithm,
This will be used to prove the identity.
Substitute

Substitute

\({\color{#009600}{x}}\ {\color{#009600}{\bm{\Large{\checkmark }}}}\)
Therefore, to the power of is equal to
Consider the following exponential and logarithmic functions. By using the Inverse Property of Logarithms, the composition of these functions results in the identity function.

This means that exponential and logarithmic functions are inverse functions. Therefore, the graphs of these functions are each other's reflection across the line

inverse graphs
Example

Modeling Battery Charges Using Logarithmic Functions

Zosia is getting everything ready for the multi-day hike in Mount Aconcagua. Because her extra battery pack is quite limited, she wants her smartphone and camera to have their batteries fully charged.

Phone and camera.png

The night before the hike, Zosia charges her smartphone and camera. She does not want to leave her devices plugged all night long because it is a waste of energy and could damage the batteries. She read in the user manuals of both devices that the charge in the batteries can be modeled by two logarithmic functions.
Here, and are the phone's and camera's charge, respectively, hours after the devices were plugged in. Zosia wants to graph these two functions on separate coordinate planes. This will help her understand at what time to unplug her devices. Help Zosia graph these functions!

Answer

Phone's Battery:

Graph of p(x), which models phone's battery

Camera's Battery:

Graph of t(x), which models tablet's battery

Hint

To graph first graph its inverse function and then reflect the curve across the line To graph make a table of values. Consider its domain first.

Solution

The logarithmic functions will be graphed one at a time.

Graphing

It is known that logarithms and exponents are inverse operations. Now, consider the natural base exponential function By the Inverse Properties of Logarithms, it can be shown that the compositions of and result in the identity function.
Therefore, and are inverse functions. This means that the graph of is a reflection across the line of the graph of Since is a natural base exponential function, the graph can be drawn by using its initial value and constant multiplier.
graphs of g(x) and p(x)

Graphing

To graph this function, a table of values will be used. Before that, however, the domain of should be determined. To do that, recall that logarithms are defined only for positive numbers. Therefore, whatever the expression is in the logarithm, it must be positive.
The domain of the function is the set of all real numbers greater than To make the table, only values in the domain will be considered.
These points can now be plotted on a coordinate plane and connected with a smooth curve.
graph of t(x)
Example

Finding Inverse Functions of Logarithmic and Exponential Functions

The expedition was a complete success! What is more, some members of the hiking team felt so inspired by Zosia’s math skills in helping them reach the summit, that they decided to practice some logarithmic functions. They believe this knowledge will also help them in their next expedition to the Amazon Rainforest.

Amazon rainforest.jpeg

Touched by their passion to learn, Zosia found a few interesting exercises about exponential and logarithmic functions in her online textbook to share with the team.

a Find the inverse function of
b Find the inverse function of

Hint
a The inverse of is a common base exponential function.
b The inverse of is a natural base logarithmic function.

Solution
a To find the inverse of the function, the first step is replacing with
Now, the and variables need to be switched.
Next, in the obtained equation, will be isolated.
DivEqn

MoveNumRight

The definition of a logarithm will be used to express the logarithmic equation as an exponential equation. Since it is a common logarithm, its base is
Finally, can be added to both sides of the equation.
AddEqn

RearrangeEqn

Rearrange equation

Substitute

To verify that the obtained function is actually the inverse of the given function, both will be graphed on the same coordinate plane. To do so, two tables of values will be made. The first table corresponds to

Next, the table that corresponds to will be constructed.

Now the points can be plotted on a coordinate plane and connected with smooth curves. Also, the line will be graphed.

function and inverse

The graphs are each other's reflection across the line Therefore, is the inverse function of

b As in Part A, to find the inverse of the function the first step is replacing with
Now, the and variables need to be switched.
Next, in the obtained equation, will be isolated.
MultEqn

The definition of a logarithm will be used to express the exponential equation as a logarithmic equation.
Finally, both sides of the equation can be divided by
DivEqn

MoveNumRight

RearrangeEqn

Rearrange equation

Substitute

To verify that the obtained function is actually the inverse of the given function, both will be graphed on the same coordinate plane. To do so, two tables of values will be made. The first table corresponds to

Next, the table that corresponds to will be constructed.

The obtained points can be plotted on a coordinate plane and connected with smooth curves. Also, the line will be graphed.

function and inverse

It can be seen that the graphs are each other's reflection across the line Therefore, is the inverse function of

Checking Our Answer
a To verify that the obtained function is actually the inverse of the given function, instead of graphing it can be checked that both composite functions and simplify to Start by calculating
Now, substitute in the above equation, and simplify as much as possible.
Substitute

▼
Simplify right-hand side
SubTerm

Subtract term

AssociativePropMult

Associative Property of Multiplication

DenomMultFracToNumber

IdPropMult

Identity Property of Multiplication

\(f(f^{\text{-} 1}(x))=x\ {\color{#009600}{\bm{\Large{\checkmark }}}}\)
Next, by following the same procedure, check that also simplifies to the identity function.
Now, substitute in the above equation, and simplify as much as possible.
Substitute

▼
Simplify right-hand side
AssociativePropMult

Associative Property of Multiplication

FracMultDenomToNumber

IdPropMult

Identity Property of Multiplication

AddTerms

Add terms

\(f^{\text{-} 1}(f(x))=x\ {\color{#009600}{\bm{\Large{\checkmark }}}}\)
Since both compositions simplify to the identity function, is the inverse function of
b As in the previous part, to check whether is the inverse function of the compositions and must both simplify to
Definition of the First Function Substitute the Second Function Simplify
$x\ {\color{#009600}{\bm{\Large{\checkmark }}}}$
$x\ {\color{#009600}{\bm{\Large{\checkmark }}}}$

Since both compositions simplify to the identity function, is the inverse function of

Closure

Horizontal and Vertical Asymptotes

In this lesson, exponential functions, logarithmic functions, and the relationship between their graphs have been discussed. One more characteristic of these graphs is that they have asymptotes.

Exponential Graphs

Every exponential graph has a horizontal asymptote. In the applet below, both graphs have the horizontal line which is the axis, as an asymptote.
exponential functions

Logarithmic Graphs

Every logarithmic graph has a vertical asymptote. In the applet below, both graphs have the vertical line which is the axis, as an asymptote.

logarithmic functions


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