Exponential and Logarithmic Functions

Graphs of Logarithmic Functions

Learning Objectives

In this section, you will:

• Identify the domain of a logarithmic function.
• Graph logarithmic functions.

In Graphs of Exponential Functions, we saw how creating a graphical representation of an exponential model gives us another layer of insight for predicting future events. How do logarithmic graphs give us insight into situations? Because every logarithmic function is the inverse function of an exponential function, we can think of every output on a logarithmic graph as the input for the corresponding inverse exponential equation. In other words, logarithms give the cause for an effect.

To illustrate, suppose we investin an account that offers an annual interest rate ofcompounded continuously. We already know that the balance in our account for any yearcan be found with the equation

But what if we wanted to know the year for any balance? We would need to create a corresponding new function by interchanging the input and the output; thus we would need to create a logarithmic model for this situation. By graphing the model, we can see the output (year) for any input (account balance). For instance, what if we wanted to know how many years it would take for our initial investment to double? (Figure) shows this point on the logarithmic graph.

In this section we will discuss the values for which a logarithmic function is defined, and then turn our attention to graphing the family of logarithmic functions.

Finding the Domain of a Logarithmic Function

Before working with graphs, we will take a look at the domain (the set of input values) for which the logarithmic function is defined.

Recall that the exponential function is defined asfor any real numberand constant where

• The domain ofis
• The range ofis

In the last section we learned that the logarithmic functionis the inverse of the exponential functionSo, as inverse functions:

• The domain ofis the range of$\left(0,\infty \right).$
• The range ofis the domain of$\left(-\infty ,\infty \right).$

Transformations of the parent functionbehave similarly to those of other functions. Just as with other parent functions, we can apply the four types of transformations—shifts, stretches, compressions, and reflections—to the parent function without loss of shape.

In Graphs of Exponential Functions we saw that certain transformations can change the range ofSimilarly, applying transformations to the parent functioncan change the domain. When finding the domain of a logarithmic function, therefore, it is important to remember that the domain consists only of positive real numbers. That is, the argument of the logarithmic function must be greater than zero.

For example, considerThis function is defined for any values ofsuch that the argument, in this case is greater than zero. To find the domain, we set up an inequality and solve for

In interval notation, the domain ofis

How To

Given a logarithmic function, identify the domain.

1. Set up an inequality showing the argument greater than zero.
2. Solve for
3. Write the domain in interval notation.

Identifying the Domain of a Logarithmic Shift

What is the domain of

The logarithmic function is defined only when the input is positive, so this function is defined whenSolving this inequality,

Try It

What is the domain of

Identifying the Domain of a Logarithmic Shift and Reflection

What is the domain of

The logarithmic function is defined only when the input is positive, so this function is defined whenSolving this inequality,

Try It

What is the domain of

Graphing Logarithmic Functions

Now that we have a feel for the set of values for which a logarithmic function is defined, we move on to graphing logarithmic functions. The family of logarithmic functions includes the parent functionalong with all its transformations: shifts, stretches, compressions, and reflections.

We begin with the parent functionBecause every logarithmic function of this form is the inverse of an exponential function with the form their graphs will be reflections of each other across the lineTo illustrate this, we can observe the relationship between the input and output values ofand its equivalentin (Figure).

Using the inputs and outputs from (Figure), we can build another table to observe the relationship between points on the graphs of the inverse functionsandSee (Figure).

As we’d expect, the x– and y-coordinates are reversed for the inverse functions. (Figure) shows the graph ofand

Observe the following from the graph:

• has a y-intercept atandhas an x– intercept at
• The domain of is the same as the range of
• The range of is the same as the domain of

Characteristics of the Graph of the Parent Function, f(x) = logb(x)

For any real numberand constant$b\ne 1,$ we can see the following characteristics in the graph of

• one-to-one function
• vertical asymptote:
• domain:
• range:
• x-intercept:and key point
• y-intercept: none
• increasing if
• decreasing if

See (Figure).

(Figure) shows how changing the baseincan affect the graphs. Observe that the graphs compress vertically as the value of the base increases. (Note: recall that the functionhas base

How To

Given a logarithmic function with the form graph the function.

1. Draw and label the vertical asymptote,
2. Plot the x-intercept,
3. Plot the key point
4. Draw a smooth curve through the points.
5. State the domain,the range,and the vertical asymptote,

Graphing a Logarithmic Function with the Form f(x) = logb(x).

GraphState the domain, range, and asymptote.

Before graphing, identify the behavior and key points for the graph.

• Sinceis greater than one, we know the function is increasing. The left tail of the graph will approach the vertical asymptote and the right tail will increase slowly without bound.
• The x-intercept is
• The key pointis on the graph.
• We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points (see (Figure)).

The domain is the range is and the vertical asymptote is[/hidden-answer]

Try It

GraphState the domain, range, and asymptote.

The domain isthe range is and the vertical asymptote is

Graphing Transformations of Logarithmic Functions

As we mentioned in the beginning of the section, transformations of logarithmic graphs behave similarly to those of other parent functions. We can shift, stretch, compress, and reflect the parent functionwithout loss of shape.

Graphing a Horizontal Shift of f(x) = logb(x)

When a constantis added to the input of the parent function the result is a horizontal shiftunits in the opposite direction of the sign onTo visualize horizontal shifts, we can observe the general graph of the parent functionand foralongside the shift left, and the shift right, See (Figure).

Horizontal Shifts of the Parent Function y = logb(x)

For any constantthe function

• shifts the parent functionleftunits if
• shifts the parent functionrightunits if
• has the vertical asymptote
• has domain
• has range

How To

Given a logarithmic function with the form graph the translation.

1. Identify the horizontal shift:
1. Ifshift the graph ofleftunits.
2. Ifshift the graph ofrightunits.
2. Draw the vertical asymptote
3. Identify three key points from the parent function. Find new coordinates for the shifted functions by subtractingfrom thecoordinate.
4. Label the three points.
5. The Domain isthe range is and the vertical asymptote is

Graphing a Horizontal Shift of the Parent Function y = logb(x)

Sketch the horizontal shiftalongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.

Since the function is we notice

ThussoThis means we will shift the functionright 2 units.

The vertical asymptote isor

Consider the three key points from the parent function,$\left(1,0\right),$and

The new coordinates are found by adding 2 to thecoordinates.

Label the points$\left(3,0\right),$and

The domain isthe range isand the vertical asymptote is

Try It

Sketch a graph ofalongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.

The domain isthe rangeand the asymptote

Graphing a Vertical Shift of y = logb(x)

When a constantis added to the parent functionthe result is a vertical shiftunits in the direction of the sign onTo visualize vertical shifts, we can observe the general graph of the parent functionalongside the shift up,and the shift down,See (Figure).

Vertical Shifts of the Parent Function y = logb(x)

For any constantthe function

• shifts the parent functionupunits if
• shifts the parent functiondownunits if
• has the vertical asymptote
• has domain
• has range

How To

Given a logarithmic function with the form graph the translation.

1. Identify the vertical shift:
• If shift the graph ofup units.
• If shift the graph ofdown units.
2. Draw the vertical asymptote
3. Identify three key points from the parent function. Find new coordinates for the shifted functions by addingto thecoordinate.
4. Label the three points.
5. The domain isthe range isand the vertical asymptote is

Graphing a Vertical Shift of the Parent Function y = logb(x)

Sketch a graph ofalongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

Since the function iswe will noticeThus

This means we will shift the functiondown 2 units.

The vertical asymptote is

Consider the three key points from the parent function,$\left(1,0\right),$and

The new coordinates are found by subtracting 2 from the y coordinates.

Label the points$\left(1,-2\right),$ and

The domain isthe range is and the vertical asymptote is

The domain isthe range isand the vertical asymptote is[/hidden-answer]

Try It

Sketch a graph ofalongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

The domain isthe range isand the vertical asymptote is

Graphing Stretches and Compressions of y = logb(x)

When the parent functionis multiplied by a constant the result is a vertical stretch or compression of the original graph. To visualize stretches and compressions, we setand observe the general graph of the parent functionalongside the vertical stretch,and the vertical compression,See (Figure).

Vertical Stretches and Compressions of the Parent Function y = logb(x)

For any constantthe function

• stretches the parent functionvertically by a factor ofif
• compresses the parent functionvertically by a factor ofif
• has the vertical asymptote
• has the x-intercept
• has domain
• has range

Given a logarithmic function with the form$a>0,$graph the translation.

1. Identify the vertical stretch or compressions:
• Ifthe graph ofis stretched by a factor ofunits.
• Ifthe graph ofis compressed by a factor ofunits.
2. Draw the vertical asymptote
3. Identify three key points from the parent function. Find new coordinates for the shifted functions by multiplying thecoordinates by
4. Label the three points.
5. The domain isthe range isand the vertical asymptote is

Graphing a Stretch or Compression of the Parent Function y = logb(x)

Sketch a graph ofalongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

[hidden-answer a=”420184″]Since the function iswe will notice

This means we will stretch the functionby a factor of 2.

The vertical asymptote is

Consider the three key points from the parent function,$\left(1,0\right),\,$and

The new coordinates are found by multiplying thecoordinates by 2.

Label the points$\left(1,0\right)\,,$ and

The domain is the range isand the vertical asymptote isSee (Figure).

The domain is the range is and the vertical asymptote is

Try It

Sketch a graph ofalongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

The domain isthe range isand the vertical asymptote is

Combining a Shift and a Stretch

Sketch a graph ofState the domain, range, and asymptote.

Remember: what happens inside parentheses happens first. First, we move the graph left 2 units, then stretch the function vertically by a factor of 5, as in (Figure). The vertical asymptote will be shifted toThe x-intercept will beThe domain will beTwo points will help give the shape of the graph:andWe choseas the x-coordinate of one point to graph because when$\,x+2=10,\,$the base of the common logarithm.

The domain isthe range isand the vertical asymptote is[/hidden-answer]

Try It

Sketch a graph of the functionState the domain, range, and asymptote.

The domain isthe range isand the vertical asymptote is

Graphing Reflections of f(x) = logb(x)

When the parent functionis multiplied bythe result is a reflection about the x-axis. When the input is multiplied bythe result is a reflection about the y-axis. To visualize reflections, we restrictand observe the general graph of the parent functionalongside the reflection about the x-axis,and the reflection about the y-axis,

Reflections of the Parent Function y = logb(x)

The function

• reflects the parent functionabout the x-axis.
• has domain, range, and vertical asymptote, which are unchanged from the parent function.

The function

• reflects the parent functionabout the y-axis.
• has domain
• has range, and vertical asymptote, which are unchanged from the parent function.

Given a logarithmic function with the parent function graph a translation.

1. Draw the vertical asymptote,
1. Draw the vertical asymptote,
1. Plot the x-intercept,
1. Plot the x-intercept,
1. Reflect the graph of the parent functionabout the x-axis.
1. Reflect the graph of the parent functionabout the y-axis.
1. Draw a smooth curve through the points.
1. Draw a smooth curve through the points.
1. State the domain, the range, and the vertical asymptote
1. State the domain, the range, and the vertical asymptote

Graphing a Reflection of a Logarithmic Function

Sketch a graph ofalongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

Before graphingidentify the behavior and key points for the graph.

• Sinceis greater than one, we know that the parent function is increasing. Since the input value is multiplied by$f\,$is a reflection of the parent graph about the y-axis. Thus,will be decreasing asmoves from negative infinity to zero, and the right tail of the graph will approach the vertical asymptote
• The x-intercept is
• We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points.

The domain isthe range isand the vertical asymptote is[/hidden-answer]

Try It

GraphState the domain, range, and asymptote.

The domain isthe range isand the vertical asymptote is

How To

Given a logarithmic equation, use a graphing calculator to approximate solutions.

1. Press [Y=]. Enter the given logarithm equation or equations as Y1= and, if needed, Y2=.
2. Press [GRAPH] to observe the graphs of the curves and use [WINDOW] to find an appropriate view of the graphs, including their point(s) of intersection.
3. To find the value of we compute the point of intersection. Press [2ND] then [CALC]. Select “intersect” and press [ENTER] three times. The point of intersection gives the value offor the point(s) of intersection.

Approximating the Solution of a Logarithmic Equation

Solvegraphically. Round to the nearest thousandth.

[hidden-answer a=”fs-id1165135193432″]Press [Y=] and enternext to Y1=. Then enternext to Y2=. For a window, use the values 0 to 5 forand –10 to 10 forPress [GRAPH]. The graphs should intersect somewhere a little to right of

For a better approximation, press [2ND] then [CALC]. Select [5: intersect] and press [ENTER] three times. The x-coordinate of the point of intersection is displayed as 1.3385297. (Your answer may be different if you use a different window or use a different value for Guess?) So, to the nearest thousandth,

Try It

Solvegraphically. Round to the nearest thousandth.

Summarizing Translations of the Logarithmic Function

Now that we have worked with each type of translation for the logarithmic function, we can summarize each in (Figure) to arrive at the general equation for translating exponential functions.

Translations of the Parent Function
Translation Form
Shift

• Horizontallyunits to the left
• Verticallyunits up
Stretch and Compress

• Stretch if
• Compression if
General equation for all translations

Translations of Logarithmic Functions

All translations of the parent logarithmic function, have the form

where the parent function,is

• shifted vertically upunits.
• shifted horizontally to the leftunits.
• stretched vertically by a factor ofif
• compressed vertically by a factor ofif
• reflected about the x-axis when

For the graph of the parent function is reflected about the y-axis.

Finding the Vertical Asymptote of a Logarithm Graph

What is the vertical asymptote of

The vertical asymptote is at

Analysis

The coefficient, the base, and the upward translation do not affect the asymptote. The shift of the curve 4 units to the left shifts the vertical asymptote to

Try It

What is the vertical asymptote of

Finding the Equation from a Graph

Find a possible equation for the common logarithmic function graphed in (Figure).

This graph has a vertical asymptote atand has been vertically reflected. We do not know yet the vertical shift or the vertical stretch. We know so far that the equation will have form:

It appears the graph passes through the pointsandSubstituting

Next, substituting in,

This gives us the equation

Analysis

We can verify this answer by comparing the function values in (Figure) with the points on the graph in (Figure).

 −1 0 1 2 3 1 0 −0.58496 −1 −1.3219 4 5 6 7 8 −1.5850 −1.8074 −2 −2.1699 −2.3219

Try It

Give the equation of the natural logarithm graphed in (Figure).

Is it possible to tell the domain and range and describe the end behavior of a function just by looking at the graph?

Yes, if we know the function is a general logarithmic function. For example, look at the graph in (Figure). The graph approaches(or thereabouts) more and more closely, sois, or is very close to, the vertical asymptote. It approaches from the right, so the domain is all points to the right,The range, as with all general logarithmic functions, is all real numbers. And we can see the end behavior because the graph goes down as it goes left and up as it goes right. The end behavior is that asand as

Access these online resources for additional instruction and practice with graphing logarithms.

Key Equations

 General Form for the Translation of the Parent Logarithmic Function

Key Concepts

• To find the domain of a logarithmic function, set up an inequality showing the argument greater than zero, and solve forSee (Figure) and (Figure)
• The graph of the parent functionhas an x-intercept atdomainrangevertical asymptoteand
• ifthe function is increasing.
• if the function is decreasing.

See (Figure).

• The equationshifts the parent functionhorizontally
• leftunits if
• rightunits if

See (Figure).

• The equationshifts the parent functionvertically
• upunits if
• downunits if

See (Figure).

• For any constant the equation
• stretches the parent functionvertically by a factor ofif
• compresses the parent functionvertically by a factor ofif

See (Figure) and (Figure).

• When the parent functionis multiplied by the result is a reflection about the x-axis. When the input is multiplied by the result is a reflection about the y-axis.
• The equationrepresents a reflection of the parent function about the x-axis.
• The equationrepresents a reflection of the parent function about the y-axis.

See (Figure).

• A graphing calculator may be used to approximate solutions to some logarithmic equations See (Figure).
• All translations of the logarithmic function can be summarized by the general equationSee (Figure).
• Given an equation with the general formwe can identify the vertical asymptotefor the transformation. See (Figure).
• Using the general equationwe can write the equation of a logarithmic function given its graph. See (Figure).

Section Exercises

Verbal

The inverse of every logarithmic function is an exponential function and vice-versa. What does this tell us about the relationship between the coordinates of the points on the graphs of each?

Since the functions are inverses, their graphs are mirror images about the lineSo for every pointon the graph of a logarithmic function, there is a corresponding pointon the graph of its inverse exponential function.

What type(s) of translation(s), if any, affect the range of a logarithmic function?

What type(s) of translation(s), if any, affect the domain of a logarithmic function?

[hidden-answer a=”917821″]Shifting the function right or left and reflecting the function about the y-axis will affect its domain.[/hidden-answer]

Consider the general logarithmic functionWhy can’tbe zero?

Does the graph of a general logarithmic function have a horizontal asymptote? Explain.

No. A horizontal asymptote would suggest a limit on the range, and the range of any logarithmic function in general form is all real numbers.

Algebraic

For the following exercises, state the domain and range of the function.

Domain:Range:

Domain:Range:

For the following exercises, state the domain and the vertical asymptote of the function.

Domain:Vertical asymptote:

Domain:Vertical asymptote:

Domain:Vertical asymptote:

For the following exercises, state the domain, vertical asymptote, and end behavior of the function.

For the following exercises, state the domain, range, and x– and y-intercepts, if they exist. If they do not exist, write DNE.

Domain:Range:Vertical asymptote:x-intercept:y-intercept: DNE

Domain:Range:Vertical asymptote:x-intercept:y-intercept: DNE

Domain:Range: Vertical asymptote: x-intercept:y-intercept: DNE

Graphical

For the following exercises, match each function in (Figure) with the letter corresponding to its graph.

B

C

For the following exercises, match each function in (Figure) with the letter corresponding to its graph.

C

For the following exercises, sketch the graphs of each pair of functions on the same axis.

and

and

and

and

For the following exercises, match each function in (Figure) with the letter corresponding to its graph.

For the following exercises, sketch the graph of the indicated function.

For the following exercises, write a logarithmic equation corresponding to the graph shown.

Useas the parent function.

Useas the parent function.

Useas the parent function.

Useas the parent function.

Technology

For the following exercises, use a graphing calculator to find approximate solutions to each equation.

Extensions

Letbe any positive real number such thatWhat mustbe equal to? Verify the result.

Explore and discuss the graphs ofandMake a conjecture based on the result.

The graphs ofandappear to be the same; Conjecture: for any positive base$\,{\mathrm{log}}_{b}\left(x\right)=-{\mathrm{log}}_{\frac{1}{b}}\left(x\right).$

Prove the conjecture made in the previous exercise.

What is the domain of the functionDiscuss the result.