# 6.4 Graphs of logarithmic functions  (Page 6/8)

 Page 6 / 8

Given a logarithmic function with the parent function $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(x\right),$ graph a translation.

1. Draw the vertical asymptote, $\text{\hspace{0.17em}}x=0.$
1. Draw the vertical asymptote, $\text{\hspace{0.17em}}x=0.$
1. Plot the x- intercept, $\text{\hspace{0.17em}}\left(1,0\right).$
1. Plot the x- intercept, $\text{\hspace{0.17em}}\left(1,0\right).$
1. Reflect the graph of the parent function $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ about the x -axis.
1. Reflect the graph of the parent function $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ about the y -axis.
1. Draw a smooth curve through the points.
1. Draw a smooth curve through the points.
1. State the domain, $\text{\hspace{0.17em}}\left(0,\infty \right),$ the range, $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ and the vertical asymptote $\text{\hspace{0.17em}}x=0.$
1. State the domain, $\text{\hspace{0.17em}}\left(-\infty ,0\right),$ the range, $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ and the vertical asymptote $\text{\hspace{0.17em}}x=0.$

## Graphing a reflection of a logarithmic function

Sketch a graph of $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{log}\left(-x\right)\text{\hspace{0.17em}}$ alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

Before graphing $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{log}\left(-x\right),$ identify the behavior and key points for the graph.

• Since $\text{\hspace{0.17em}}b=10\text{\hspace{0.17em}}$ is greater than one, we know that the parent function is increasing. Since the input value is multiplied by $\text{\hspace{0.17em}}-1,$ $f\text{\hspace{0.17em}}$ is a reflection of the parent graph about the y- axis. Thus, $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{log}\left(-x\right)\text{\hspace{0.17em}}$ will be decreasing as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ moves from negative infinity to zero, and the right tail of the graph will approach the vertical asymptote $\text{\hspace{0.17em}}x=0.\text{\hspace{0.17em}}$
• The x -intercept is $\text{\hspace{0.17em}}\left(-1,0\right).$
• We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points.

The domain is $\text{\hspace{0.17em}}\left(-\infty ,0\right),$ the range is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ and the vertical asymptote is $\text{\hspace{0.17em}}x=0.$

Graph $\text{\hspace{0.17em}}f\left(x\right)=-\mathrm{log}\left(-x\right).\text{\hspace{0.17em}}$ State the domain, range, and asymptote.

The domain is $\text{\hspace{0.17em}}\left(-\infty ,0\right),$ the range is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ and the vertical asymptote is $\text{\hspace{0.17em}}x=0.$

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

1. Press [Y=] . Enter the given logarithm equation or equations as Y 1 = and, if needed, Y 2 = .
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 $\text{\hspace{0.17em}}x,$ we compute the point of intersection. Press [2ND] then [CALC] . Select “intersect” and press [ENTER] three times. The point of intersection gives the value of $\text{\hspace{0.17em}}x,$ for the point(s) of intersection.

## Approximating the solution of a logarithmic equation

Solve $\text{\hspace{0.17em}}4\mathrm{ln}\left(x\right)+1=-2\mathrm{ln}\left(x-1\right)\text{\hspace{0.17em}}$ graphically. Round to the nearest thousandth.

Press [Y=] and enter $\text{\hspace{0.17em}}4\mathrm{ln}\left(x\right)+1\text{\hspace{0.17em}}$ next to Y 1 =. Then enter $\text{\hspace{0.17em}}-2\mathrm{ln}\left(x-1\right)\text{\hspace{0.17em}}$ next to Y 2 = . For a window, use the values 0 to 5 for $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and –10 to 10 for $\text{\hspace{0.17em}}y.\text{\hspace{0.17em}}$ Press [GRAPH] . The graphs should intersect somewhere a little to right of $\text{\hspace{0.17em}}x=1.$

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, $\text{\hspace{0.17em}}x\approx 1.339.$

Solve $\text{\hspace{0.17em}}5\mathrm{log}\left(x+2\right)=4-\mathrm{log}\left(x\right)\text{\hspace{0.17em}}$ graphically. Round to the nearest thousandth.

$x\approx 3.049$

## 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 [link] to arrive at the general equation for translating exponential functions.

#### Questions & Answers

bsc F. y algebra and trigonometry pepper 2
Aditi Reply
given that x= 3/5 find sin 3x
Adamu Reply
4
DB
remove any signs and collect terms of -2(8a-3b-c)
Joeval Reply
-16a+6b+2c
Will
is that a real answer
Joeval
(x2-2x+8)-4(x2-3x+5)
Ayush Reply
sorry
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
(X2-2X+8)-4(X2-3X+5)=0 ?
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
Y
master
X2-2X+8-4X2+12X-20=0 (X2-4X2)+(-2X+12X)+(-20+8)= 0 -3X2+10X-12=0 3X2-10X+12=0 Use quadratic formula To find the answer answer (5±Root11i)/3
master
Soo sorry (5±Root11* i)/3
master
x2-2x+8-4x2+12x-20 x2-4x2-2x+12x+8-20 -3x2+10x-12 now you can find the answer using quadratic
Mukhtar
explain and give four example of hyperbolic function
Lukman Reply
What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?
Racelle Reply
y/y+10
Mr
Find nth derivative of eax sin (bx + c).
Anurag Reply
Find area common to the parabola y2 = 4ax and x2 = 4ay.
Anurag
A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden
Jhovie Reply
to find the length I divide the area by the wide wich means 1125ft/25ft=45
Miranda
thanks
Jhovie
What do you call a relation where each element in the domain is related to only one value in the range by some rules?
Charmaine Reply
A banana.
Yaona
given 4cot thither +3=0and 0°<thither <180° use a sketch to determine the value of the following a)cos thither
Snalo Reply
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Mark Reply
nothing up todat yet
Miranda
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jai
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jai
Miranda Drice
jai
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jai
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Miranda
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jai
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Miranda
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Propessor Reply
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
sita Reply
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
hi
jai
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jai
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Propessor
welcome
jai
What is algebra
Pearl Reply
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
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Jeffrey
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Miranda
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Jeffrey
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Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
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Miranda
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Miranda
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Jeffrey
how about you? what grade are you now?
Jeffrey
I'm going to 11grade
Miranda
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Miranda
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Steve
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Steve
I don't know why. But Im trying to like it.
Jeffrey
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Jeffrey
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Miranda
what is the solution of the given equation?
Nelson Reply
which equation
Miranda
I dont know. lol
Jeffrey
please where is the equation
Miranda
ask nelson. lol
Jeffrey

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