Extra Graphing Review Graph Each and Find the X Intercepts F(X) = Log4 X

Learning Objectives

In this section, you volition:

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

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

To illustrate, suppose we invest $\text{\$}2500$ in an account that offers an annual interest rate of $\mathrm{v}\%,$ compounded continuously. We already know that the remainder in our account for whatsoever yr $t$ can exist institute with the equation $A=2500{\mathrm{due\; east}}^{0.05t}.$

Only what if we wanted to know the year for any residuum? We would demand 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. Past graphing the model, we tin meet the output (year) for any input (business relationship residuum). For case, what if nosotros wanted to know how many years information technology would take for our initial investment to double? Effigy ane shows this indicate on the logarithmic graph.

Figure one

In this section nosotros volition talk over the values for which a logarithmic office is defined, and and then plough our attention to graphing the family of logarithmic functions.

Finding the Domain of a Logarithmic Role

Before working with graphs, nosotros will take a await at the domain (the gear up of input values) for which the logarithmic function is defined.

Recall that the exponential function is defined as $y=b^{\mathrm{10}}$ for whatever existent number $\mathrm{ten}$ and abiding $b>0,$ $b\ne 1,$ where

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

In the last section we learned that the logarithmic function $y={\log }_b\left(x\right)$ is the inverse of the exponential function $y=b^x.$ So, every bit inverse functions:

• The domain of $y={\log }_b\left(x\right)$ is the range of $y=b^{\mathrm{10}}:$ $\left(0,\infty \right).$
• The range of $y={\log }_b\left(\mathrm{ten}\right)$ is the domain of $y=b^{\mathrm{ten}}:$ $\left(-\infty ,\infty \right).$

Transformations of the parent function $y={\log }_b\left(x\right)$ behave similarly to those of other functions. Merely as with other parent functions, we tin can apply the 4 types of transformations—shifts, stretches, compressions, and reflections—to the parent function without loss of shape.

In Graphs of Exponential Functions we saw that sure transformations can change the range of $y=b^x.$ Similarly, applying transformations to the parent function $y={\log }_b\left(\mathrm{10}\right)$ tin can change the domain. When finding the domain of a logarithmic office, therefore, it is of import to retrieve that the domain consists just of positive real numbers. That is, the statement of the logarithmic function must be greater than goose egg.

For example, consider $f(x)={\log }_4\left(2x-\mathrm{three}\right).$ This part is defined for whatever values of $x$ such that the statement, in this instance $\mathrm{ii}\mathrm{10}-3,$ is greater than zero. To discover the domain, we set up an inequality and solve for $\mathrm{10}:$

$$\begin{array}{ll}2x-3>0\hfill & \text{Show the argument greater than zip}.\hfill \\ 2x>\mathrm{three}\hfill & \text{Add 3}.\hfill \\ x>1.5\begin{array}{cccc}& & & \end{array}\hfill & \text{Divide past 2}.\hfill \end{array}$$

In interval notation, the domain of $f(x)={\log }_4\left(2x-3\right)$ is $\left(\mathrm{i.5},\infty \right).$

How To

Given a logarithmic function, identify the domain.

1. Set upward an inequality showing the statement greater than zero.
2. Solve for $x.$
3. Write the domain in interval notation.

Instance 1

Identifying the Domain of a Logarithmic Shift

What is the domain of $f(\mathrm{ten})={\log }_2(x+3)?$

Effort Information technology #1

What is the domain of $f(x)={\log }_{\mathrm{five}}(\mathrm{10}-\mathrm{ii})+\mathrm{i}?$

Example two

Identifying the Domain of a Logarithmic Shift and Reflection

What is the domain of $f(\mathrm{ten})=\log (5-2x)?$

Try It #ii

What is the domain of $f(\mathrm{ten})=\log (\mathrm{ten}-\mathrm{v})+2?$

Graphing Logarithmic Functions

Now that nosotros have a feel for the set of values for which a logarithmic function is defined, we motility on to graphing logarithmic functions. The family unit of logarithmic functions includes the parent function $y={\log }_b\left(x\right)$ forth with all its transformations: shifts, stretches, compressions, and reflections.

We brainstorm with the parent function $y={\log }_b\left(\mathrm{ten}\right).$ Because every logarithmic office of this course is the inverse of an exponential function with the form $y=b^x,$ their graphs volition exist reflections of each other beyond the line $y=x.$ To illustrate this, we can discover the relationship between the input and output values of $y={\mathrm{ii}}^x$ and its equivalent $x={\log }_{\mathrm{two}}(y)$ in Table ane.

$x$	$-3$	$-2$	$-1$	$0$	$1$	$\mathrm{two}$	$3$
$2^{\mathrm{ten}}=y$	$\frac{1}{\mathrm{viii}}$	$\frac{\mathrm{i}}{\mathrm{four}}$	$\frac{\mathrm{i}}{\mathrm{ii}}$	$1$	$2$	$4$	$8$
${\log }_2\left(y\right)=x$	$-3$	$-2$	$-\mathrm{ane}$	$0$	$1$	$2$	$3$

Table ane

Using the inputs and outputs from Table 1, we can build another table to find the relationship between points on the graphs of the inverse functions $f(\mathrm{ten})=2^x$ and $\mathrm{1000}(x)={\log }_{\mathrm{two}}(x).$ Meet Table ii.

$f(\mathrm{ten})=2^x$	$\left(-3,\frac{1}{8}\right)$	$\left(-2,\frac{1}{4}\right)$	$\left(-1,\frac{1}{2}\right)$	$\left(0,1\right)$	$\left(1,\mathrm{ii}\right)$	$\left(\mathrm{two},\mathrm{four}\right)$	$\left(\mathrm{iii},8\right)$
$\mathrm{chiliad}(x)={\log }_2\left(x\right)$	$\left(\frac{1}{\mathrm{eight}},-\mathrm{three}\right)$	$\left(\frac{1}{\mathrm{iv}},-\mathrm{two}\right)$	$\left(\frac{1}{\mathrm{ii}},-\mathrm{one}\right)$	$\left(\mathrm{ane},0\right)$	$\left(2,1\right)$	$\left(4,2\right)$	$\left(\mathrm{eight},3\right)$

Table 2

Equally we'd expect, the x- and y-coordinates are reversed for the inverse functions. Figure ii shows the graph of $f$ and $\mathrm{thou}.$

Figure two Detect that the graphs of $f\left(x\right)=2^x$ and $k\left(x\right)={\log }_2\left(x\right)$ are reflections about the line $y=\mathrm{ten}.$

Discover the post-obit from the graph:

• $f(\mathrm{ten})={\mathrm{ii}}^x$ has a y-intercept at $(0,1)$ and $g(x)={\log }_2\left(x\right)$ has an ten- intercept at $(\mathrm{i},0).$
• The domain of $f(x)=2^{\mathrm{ten}},$ $\left(-\infty ,\infty \right),$ is the same equally the range of $\mathrm{grand}(\mathrm{ten})={\log }_{\mathrm{two}}\left(x\right).$
• The range of $f(\mathrm{10})={\mathrm{two}}^x,$ $\left(0,\infty \right),$ is the aforementioned equally the domain of $g(x)={\log }_2\left(x\right).$

Characteristics of the Graph of the Parent Function, $f(\mathrm{ten})={\log }_b\left(x\right):$

For whatsoever existent number $x$ and constant $b>0,$ $b\ne 1,$ nosotros can come across the following characteristics in the graph of $f(x)={\log }_b\left(x\right):$

• 1-to-one function
• vertical asymptote: $x=0$
• domain: $(0,\infty )$
• range: $\left(-\infty ,\infty \right)$
• 10-intercept: $(1,0)$ and key point $(b,\mathrm{ane})$
• y-intercept: none
• increasing if $b>1$
• decreasing if $0<b<1$

See Figure 3.

Figure 3

Figure 4 shows how changing the base $b$ in $f(x)={\log }_b\left(x\right)$ can affect the graphs. Observe that the graphs compress vertically every bit the value of the base increases. (Annotation: recall that the function $\ln \left(x\right)$ has base $\mathrm{eastward}\approx \text{ii}.\text{718.)}$

Figure 4 The graphs of 3 logarithmic functions with different bases, all greater than 1.

How To

Given a logarithmic part with the form $f(x)={\log }_b\left(\mathrm{10}\right),$ graph the function.

1. Draw and characterization the vertical asymptote, $x=0.$
2. Plot the x-intercept, $\left(1,0\right).$
3. Plot the key point $\left(b,\mathrm{one}\right).$
4. Draw a smooth bend through the points.
5. Country the domain, $\left(0,\infty \right),$ the range, $\left(-\infty ,\infty \right),$ and the vertical asymptote, $x=0.$

Instance three

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

Graph $f(x)={\log }_{\mathrm{v}}\left(x\right).$ Country the domain, range, and asymptote.

Try It #3

Graph $f(x)={\log }_{\frac{1}{5}}(x).$ Land the domain, range, and asymptote.

Graphing Transformations of Logarithmic Functions

Every bit we mentioned in the beginning of the section, transformations of logarithmic graphs acquit similarly to those of other parent functions. We can shift, stretch, compress, and reflect the parent function $y={\log }_b\left(\mathrm{ten}\right)$ without loss of shape.

Graphing a Horizontal Shift of f(x) = log _b (ten)

When a constant $c$ is added to the input of the parent function $f(x)=\mathrm{fifty}o{\mathrm{grand}}_b(x),$ the result is a horizontal shift $c$ units in the opposite direction of the sign on $c.$ To visualize horizontal shifts, we can observe the general graph of the parent function $f(\mathrm{ten})={\log }_b\left(\mathrm{ten}\right)$ and for $c>0$ alongside the shift left, $g(\mathrm{ten})={\log }_b\left(x+c\right),$ and the shift right, $h(\mathrm{10})={\log }_b\left(x-c\right).$ See Effigy 6.

Effigy half dozen

Horizontal Shifts of the Parent Function $f(x)={\log }_b\left(x\right)$

For whatsoever constant $c,$ the function $f(\mathrm{10})={\log }_b\left(x+c\right)$

• shifts the parent function $y={\log }_b\left(\mathrm{10}\right)$ left $c$ units if $c>0.$
• shifts the parent function $y={\log }_b\left(x\right)$ right $c$ units if $c<0.$
• has the vertical asymptote $\mathrm{10}=-c.$
• has domain $\left(-c,\infty \right).$
• has range $\left(-\infty ,\infty \right).$

How To

Given a logarithmic function with the form $f(x)={\log }_b\left(x+c\right),$ graph the translation.

1. Identify the horizontal shift:
   1. If $c>0,$ shift the graph of $f(\mathrm{ten})={\log }_b\left(\mathrm{ten}\right)$ left $c$ units.
   2. If $c<0,$ shift the graph of $f(\mathrm{10})={\log }_b\left(\mathrm{10}\right)$ correct $c$ units.
2. Draw the vertical asymptote $x=-c.$
3. Identify 3 key points from the parent function. Discover new coordinates for the shifted functions past subtracting $c$ from the $x$ coordinate.
4. Characterization the iii points.
5. The Domain is $\left(-c,\infty \right),$ the range is $\left(-\infty ,\infty \right),$ and the vertical asymptote is $x=-c.$

Instance 4

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

Sketch the horizontal shift $f(x)={\log }_{\mathrm{iii}}(x-2)$ alongside its parent function. Include the fundamental points and asymptotes on the graph. State the domain, range, and asymptote.

Endeavour It #four

Sketch a graph of $f(x)={\log }_{\mathrm{iii}}(x+\mathrm{iv})$ alongside its parent part. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.

Graphing a Vertical Shift of y = log _b (x)

When a constant $d$ is added to the parent function $f(\mathrm{ten})={\log }_b\left(\mathrm{10}\right),$ the upshot is a vertical shift $d$ units in the direction of the sign on $d.$ To visualize vertical shifts, nosotros can find the general graph of the parent function $f(\mathrm{ten})={\log }_b\left(\mathrm{10}\right)$ alongside the shift up, $\mathrm{thou}(x)={\log }_b\left(x\right)+d$ and the shift down, $h(\mathrm{10})={\log }_b\left(\mathrm{ten}\right)-d.$ See Figure 8.

Figure 8

Vertical Shifts of the Parent Function $y={\log }_b\left(\mathrm{ten}\right)$

For any constant $d,$ the role $f(x)={\log }_b\left(\mathrm{ten}\right)+d$

• shifts the parent function $y={\log }_b\left(x\right)$ upwards $d$ units if $d>0.$
• shifts the parent office $y={\log }_b\left(x\right)$ down $d$ units if $d<0.$
• has the vertical asymptote $x=0.$
• has domain $\left(0,\infty \right).$
• has range $\left(-\infty ,\infty \right).$

How To

Given a logarithmic function with the grade $f(x)={\log }_b\left(x\right)+d,$ graph the translation.

1. Place the vertical shift:
   • If $d>0,$ shift the graph of $f(x)={\log }_b\left(x\right)$ up $d$ units.
   • If $d<0,$ shift the graph of $f(x)={\log }_b\left(x\right)$ down $d$ units.
2. Draw the vertical asymptote $x=0.$
3. Identify three key points from the parent role. Find new coordinates for the shifted functions by calculation $d$ to the $y$ coordinate.
4. Label the 3 points.
5. The domain is $\left(\mathrm{0,}\infty \right),$ the range is $\left(-\infty ,\infty \right),$ and the vertical asymptote is $x=0.$

Example 5

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

Sketch a graph of $f(x)={\log }_3(x)-\mathrm{two}$ alongside its parent function. Include the key points and asymptote on the graph. Country the domain, range, and asymptote.

Try It #5

Sketch a graph of $f(x)={\log }_2(\mathrm{ten})+2$ alongside its parent function. Include the cardinal points and asymptote on the graph. State the domain, range, and asymptote.

Graphing Stretches and Compressions of y = log _b (x)

When the parent office $f(\mathrm{ten})={\log }_b\left(x\right)$ is multiplied past a constant $a>0,$ the issue is a vertical stretch or pinch of the original graph. To visualize stretches and compressions, we prepare $a>1$ and detect the general graph of the parent function $f(\mathrm{10})={\log }_b\left(x\right)$ aslope the vertical stretch, $g(\mathrm{10})=a{\log }_b\left(x\right)$ and the vertical compression, $h(\mathrm{10})=\frac{1}{a}{\log }_b\left(x\right).$ See Figure 10.

Figure 10

Vertical Stretches and Compressions of the Parent Function $y={\log }_b\left(x\right)$

For whatever constant $a>\mathrm{i},$ the role $f(x)=a{\log }_b\left(x\right)$

• stretches the parent function $y={\log }_b\left(x\right)$ vertically by a factor of $a$ if $a>1.$
• compresses the parent function $y={\log }_b\left(\mathrm{ten}\right)$ vertically past a factor of $a$ if $0<a<1.$
• has the vertical asymptote $x=0.$
• has the 10-intercept $\left(1,0\right).$
• has domain $\left(0,\infty \right).$
• has range $\left(-\infty ,\infty \right).$

How To

Given a logarithmic function with the class $f(x)=a{\log }_b\left(x\right),$ $a>0,$ graph the translation.

1. Identify the vertical stretch or compressions:
   • If $\left|a\right|>1,$ the graph of $f(\mathrm{ten})={\log }_b\left(\mathrm{ten}\right)$ is stretched by a factor of $a$ units.
   • If $\left|a\right|<1,$ the graph of $f(\mathrm{10})={\log }_b\left(x\right)$ is compressed by a factor of $a$ units.
2. Draw the vertical asymptote $\mathrm{ten}=0.$
3. Identify iii key points from the parent part. Find new coordinates for the shifted functions by multiplying the $y$ coordinates by $a.$
4. Label the three points.
5. The domain is $\left(0,\infty \right),$ the range is $\left(-\infty ,\infty \right),$ and the vertical asymptote is $\mathrm{10}=0.$

Case 6

Graphing a Stretch or Compression of the Parent Office y = log _b (x)

Sketch a graph of $f(x)=2{\log }_4(x)$ alongside its parent function. Include the key points and asymptote on the graph. Land the domain, range, and asymptote.

Try It #6

Sketch a graph of $f(\mathrm{ten})=\frac{\mathrm{one}}{2}{\log }_4(x)$ alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

Example seven

Combining a Shift and a Stretch

Sketch a graph of $f(x)=5\log (x+2).$ Country the domain, range, and asymptote.

Try It #7

Sketch a graph of the function $f(x)=\mathrm{three}\log (x-2)+\mathrm{i.}$ State the domain, range, and asymptote.

Graphing Reflections of f(x) = log _b (ten)

When the parent function $f(x)={\log }_b\left(\mathrm{10}\right)$ is multiplied by $\mathrm{-i},$ the effect is a reflection near the x-axis. When the input is multiplied by $\mathrm{-ane},$ the consequence is a reflection about the y-axis. To visualize reflections, we restrict $b>\mathrm{ane,}$ and observe the full general graph of the parent function $f(x)={\log }_b\left(\mathrm{10}\right)$ alongside the reflection near the x-axis, $g(x)={\mathrm{-log}}_b\left(\mathrm{ten}\right)$ and the reflection about the y-axis, $h(\mathrm{10})={\log }_b\left(-x\right).$

Effigy 13

Reflections of the Parent Function $y={\log }_b\left(x\right)$

The function $f(x)={\mathrm{-log}}_b\left(x\right)$

• reflects the parent office $y={\log }_b\left(x\right)$ nigh the x-axis.
• has domain, $\left(0,\infty \right),$ range, $\left(-\infty ,\infty \right),$ and vertical asymptote, $x=0,$ which are unchanged from the parent office.

The function $f(x)={\log }_b\left(-\mathrm{ten}\right)$

• reflects the parent function $y={\log }_b\left(x\right)$ about the y-centrality.
• has domain $\left(-\infty ,0\right).$
• has range, $\left(-\infty ,\infty \right),$ and vertical asymptote, $\mathrm{10}=0,$ which are unchanged from the parent part.

How To

Given a logarithmic function with the parent function $f(x)={\log }_b\left(\mathrm{10}\right),$ graph a translation.

$\text{If}f(x)=-{\log }_b(x)$	$\text{If}f(\mathrm{10})={\log }_b(-x)$
i. Draw the vertical asymptote, $x=0.$	1. Depict the vertical asymptote, $x=0.$
ii. Plot the x-intercept, $\left(1,0\right).$	2. Plot the x-intercept, $\left(\mathrm{one},0\right).$
3. Reverberate the graph of the parent function $f(\mathrm{10})={\log }_b\left(x\right)$ about the x-axis.	iii. Reverberate the graph of the parent function $f(\mathrm{ten})={\log }_b\left(x\right)$ about the y-centrality.
4. Draw a smooth curve through the points.	4. Draw a shine curve through the points.
five. State the domain, (0, ∞), the range, (−∞, ∞), and the vertical asymptote $\mathrm{10}=0$.	five. State the domain, (−∞, 0) the range, (−∞, ∞) and the vertical asymptote $x=0.$

Table 3

Example 8

Graphing a Reflection of a Logarithmic Function

Sketch a graph of $f(\mathrm{10})=\log (-\mathrm{ten})$ alongside its parent function. Include the central points and asymptote on the graph. State the domain, range, and asymptote.

Attempt Information technology #viii

Graph $f(x)=-\log (-\mathrm{ten}).$ Country the domain, range, and asymptote.

How To

Given a logarithmic equation, employ a graphing calculator to guess solutions.

1. Press [Y=]. Enter the given logarithm equation or equations equally Y_ane= 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(south) of intersection.
3. To find the value of $x,$ we compute the indicate of intersection. Press [2ND] then [CALC]. Select "intersect" and printing [ENTER] iii times. The betoken of intersection gives the value of $\mathrm{ten},$ for the point(south) of intersection.

Instance 9

Approximating the Solution of a Logarithmic Equation

Solve $\mathrm{four}\ln \left(x\right)+\mathrm{i}=-2\ln \left(x-1\right)$ graphically. Circular to the nearest thousandth.

Endeavour It #9

Solve $5\log \left(x+\mathrm{ii}\right)=\mathrm{iv}-\log \left(\mathrm{10}\right)$ graphically. Circular to the nearest thousandth.

Summarizing Translations of the Logarithmic Role

Now that we have worked with each blazon of translation for the logarithmic role, nosotros tin summarize each in Table 4 to arrive at the full general equation for translating exponential functions.

Translations of the Parent Function $y={\log }_b\left(x\right)$
Translation	Course
Shift Horizontally $c$ units to the left Vertically $d$ units up	$y={\log }_b\left(x+c\right)+d$
Stretch and Compress Stretch if $\left|a\right|>\mathrm{one}$ Pinch if $\left|a\right|<1$	$y=a{\log }_b\left(\mathrm{ten}\right)$
Reflect nigh the x-axis	$y=-{\log }_b\left(x\right)$
Reflect nigh the y-axis	$y={\log }_b\left(-\mathrm{10}\right)$
General equation for all translations	$y=a{\log }_b(x+c)+d$

Tabular array 4

Translations of Logarithmic Functions

All translations of the parent logarithmic part, $y={\log }_b\left(x\right),$ take the form

$$f(x)=a{\log }_b\left(x+c\right)+d$$

where the parent function, $y={\log }_b\left(x\right),b>\mathrm{one},$ is

• shifted vertically upwardly $d$ units.
• shifted horizontally to the left $c$ units.
• stretched vertically by a factor of $\left|a\right|$ if $\left|a\right|>0.$
• compressed vertically past a cistron of $\left|a\right|$ if $0<\left|a\right|<1.$
• reflected about the x-axis when $a<0.$

For $f\left(\mathrm{10}\right)=\log \left(-x\right),$ the graph of the parent function is reflected well-nigh the y-axis.

Instance 10

Finding the Vertical Asymptote of a Logarithm Graph

What is the vertical asymptote of $f(\mathrm{ten})=\mathrm{-ii}{\log }_{\mathrm{iii}}(x+4)+5?$

Analysis

The coefficient, the base, and the upward translation exercise not affect the asymptote. The shift of the curve 4 units to the left shifts the vertical asymptote to $x=\mathrm{-4.}$

Endeavour It #10

What is the vertical asymptote of $f(x)=3+\ln (\mathrm{ten}-1)?$

Example eleven

Finding the Equation from a Graph

Find a possible equation for the common logarithmic part graphed in Effigy 15.

Figure 15

Analysis

We tin can verify this reply by comparing the office values in Table v with the points on the graph in Effigy 15.

$x$	−ane	0	ane	2	3
$f(\mathrm{10})$	i	0	−0.58496	−1	−i.3219
$x$	4	5	6	7	8
$f(x)$	−one.5850	−1.8074	−2	−2.1699	−two.3219

Table 5

Endeavor It #11

Give the equation of the natural logarithm graphed in Effigy 16.

Figure xvi

Q&A

Is information technology possible to tell the domain and range and draw the end behavior of a function just by looking at the graph?

Yeah, if nosotros know the part is a general logarithmic part. For example, look at the graph in Figure 16. The graph approaches $x=\mathrm{-3}$ (or thereabouts) more and more closely, and then $x=\mathrm{-3}$ is, or is very close to, the vertical asymptote. Information technology approaches from the correct, so the domain is all points to the correct, $\left\{x\right|x>\mathrm{-three}\}.$ The range, as with all general logarithmic functions, is all existent numbers. And we tin see the end behavior because the graph goes down as it goes left and upward as information technology goes right. The end behavior is that as $x\to -3^{+},f(x)\to -\infty$ and as $x\to \infty ,f(x)\to \infty .$

6.4 Section Exercises

Verbal

one .

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

2 .

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

three .

What blazon(southward) of translation(s), if any, affect the domain of a logarithmic office?

4 .

Consider the full general logarithmic function $f(x)={\log }_b\left(\mathrm{10}\right).$ Why tin can't $\mathrm{10}$ be zero?

five .

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

Algebraic

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

6 .

$f(x)={\log }_{\mathrm{three}}\left(x+4\right)$

7 .

$h(x)=\ln \left(\frac{\mathrm{i}}{2}-\mathrm{ten}\right)$

eight .

$\mathrm{grand}(x)={\log }_{\mathrm{five}}\left(\mathrm{ii}\mathrm{ten}+9\right)-2$

9 .

$h(\mathrm{ten})=\ln \left(4x+17\right)-5$

10 .

$f(x)={\log }_2\left(12-3x\right)-\mathrm{three}$

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

11 .

$f(\mathrm{10})={\log }_b(x-\mathrm{v})$

12 .

$g(\mathrm{ten})=\ln (3-x)$

13 .

$f(x)=\log (3x+\mathrm{i})$

fourteen .

$f(x)=\mathrm{iii}\log (-x)+2$

xv .

$g(x)=-\ln (\mathrm{three}x+9)-7$

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

16 .

$f(x)=\ln \left(2-\mathrm{ten}\right)$

17 .

$f(\mathrm{ten})=\log \left(x-\frac{3}{\mathrm{seven}}\right)$

18 .

$h(x)=-\log \left(3x-4\right)+\mathrm{iii}$

19 .

$\mathrm{1000}(x)=\ln \left(2\mathrm{10}+6\right)-\mathrm{five}$

20 .

$f(x)={\log }_3\left(\mathrm{fifteen}-5x\right)+6$

For the following exercises, state the domain, range, and x- and y-intercepts, if they exist. If they exercise non exist, write DNE.

21 .

$h(x)={\log }_{\mathrm{four}}\left(\mathrm{ten}-\mathrm{ane}\right)+1$

22 .

$f(x)=\log \left(5x+10\right)+3$

23 .

$g(x)=\ln \left(-x\right)-2$

24 .

$f(\mathrm{ten})={\log }_2\left(\mathrm{ten}+\mathrm{ii}\right)-\mathrm{v}$

25 .

$h(\mathrm{10})=3\ln \left(\mathrm{ten}\right)-9$

Graphical

For the following exercises, match each part in Effigy 17 with the letter of the alphabet respective to its graph.

Figure 17

26 .

$d(x)=\log \left(x\right)$

27 .

$f(x)=\ln (x)$

28 .

$\mathrm{chiliad}(x)={\log }_2\left(x\right)$

29 .

$h(x)={\log }_5\left(x\right)$

30 .

$j(x)={\log }_{25}\left(x\right)$

For the following exercises, match each office in Effigy 18 with the letter of the alphabet respective to its graph.

Figure 18

31 .

$f(x)={\log }_{\frac{\mathrm{one}}{\mathrm{three}}}\left(x\right)$

32 .

$\mathrm{thousand}(x)={\log }_2\left(\mathrm{ten}\right)$

33 .

$h(x)={\log }_{\frac{3}{4}}\left(x\right)$

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

34 .

$f(x)=\log (x)$ and $\mathrm{thousand}(x)={10}^{\mathrm{ten}}$

35 .

$f(\mathrm{ten})=\log (\mathrm{10})$ and $k(\mathrm{ten})={\log }_{\frac{\mathrm{i}}{2}}(x)$

36 .

$f(x)={\log }_4(x)$ and $g(\mathrm{10})=\ln (x)$

37 .

$f(\mathrm{10})=e^{\mathrm{ten}}$ and $\mathrm{thousand}(\mathrm{ten})=\ln (x)$

For the following exercises, friction match each function in Figure 19 with the letter corresponding to its graph.

Effigy nineteen

38 .

$f(x)={\log }_4\left(-\mathrm{10}+\mathrm{two}\right)$

39 .

$g(x)=-{\log }_4\left(\mathrm{10}+2\right)$

twoscore .

$h(x)={\log }_{\mathrm{iv}}\left(x+2\right)$

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

41 .

$f(x)={\log }_2(x+2)$

42 .

$f(x)=2\log (x)$

43 .

$f(\mathrm{ten})=\ln (-x)$

44 .

$m(x)=\log \left(4x+\mathrm{sixteen}\right)+\mathrm{iv}$

45 .

$g(x)=\log \left(6-3\mathrm{ten}\right)+\mathrm{ane}$

46 .

$h(x)=-\frac{\mathrm{ane}}{2}\ln \left(\mathrm{ten}+1\right)-\mathrm{iii}$

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

47 .

Apply $y={\log }_2(x)$ as the parent role.

48 .

Utilise $f(x)={\log }_3(x)$ as the parent office.

49 .

Use $f(\mathrm{10})={\log }_4(x)$ as the parent function.

fifty .

Use $f(x)={\log }_{\mathrm{v}}(\mathrm{10})$ as the parent function.

Engineering science

For the following exercises, use a graphing estimator to notice gauge solutions to each equation.

51 .

$\log \left(x-1\right)+2=\ln \left(x-1\right)+\mathrm{ii}$

52 .

$\log \left(\mathrm{two}x-3\right)+2=-\log \left(2x-3\right)+5$

53 .

$\ln \left(\mathrm{ten}-2\right)=-\ln \left(\mathrm{ten}+1\right)$

54 .

$2\ln \left(5\mathrm{10}+1\right)=\frac{1}{\mathrm{ii}}\ln \left(-5x\right)+1$

55 .

$\frac{\mathrm{one}}{3}\log \left(1-\mathrm{ten}\right)=\log \left(\mathrm{10}+\mathrm{one}\right)+\frac{1}{\mathrm{three}}$

Extensions

56 .

Allow $b$ be whatsoever positive real number such that $b\ne 1.$ What must ${\log }_{b}1$ be equal to? Verify the result.

57 .

Explore and discuss the graphs of $f(x)={\log }_{\frac{1}{\mathrm{ii}}}\left(x\right)$ and $\mathrm{chiliad}(x)=-{\log }_2\left(x\right).$ Brand a conjecture based on the result.

58 .

Evidence the conjecture made in the previous practise.

59 .

What is the domain of the part $f(\mathrm{10})=\ln \left(\frac{\mathrm{ten}+\mathrm{ii}}{x-4}\right)?$ Discuss the upshot.

60 .

Use properties of exponents to observe the 10-intercepts of the function $f(x)=\log \left({\mathrm{ten}}^{\mathrm{two}}+\mathrm{four}\mathrm{ten}+4\right)$ algebraically. Evidence the steps for solving, and then verify the result past graphing the role.

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Source: https://openstax.org/books/college-algebra-2e/pages/6-4-graphs-of-logarithmic-functions

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