# 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.

f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
proof
AUSTINE
sebd me some questions about anything ill solve for yall
how to solve x²=2x+8 factorization?
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
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Amit
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Dorbor
well
Biswajit
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Gaurav
Find the possible value of 8.5 using moivre's theorem
which of these functions is not uniformly cintinuous on (0, 1)? sinx
which of these functions is not uniformly continuous on 0,1
solve this equation by completing the square 3x-4x-7=0
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
new member
siame
what is trigonometry
deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
Thomas
solve for me this equational y=2-x
what are you solving for
Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
then I got x=-2
Rubben
it will b -y+2=x
Alex
goodness. I'm sorry. I will let Alex take the wheel.
Nikki
ouky thanks braa
Rubben
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Rubben
how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
More example of algebra and trigo
What is Indices
If one side only of a triangle is given is it possible to solve for the unkown two sides?
cool
Rubben
kya
Khushnama
please I need help in maths
Okey tell me, what's your problem is?
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