6.4 Graphs of logarithmic functions (Page 3/8)

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How To

Given a logarithmic function with the form $f (x) = \log_{b} (x),$ graph the function.

Draw and label the vertical asymptote, $x = 0.$
Plot the x- intercept, $(1, 0) .$
Plot the key point $(b, 1) .$
Draw a smooth curve through the points.
State the domain, $(0, \infty),$ the range, $(- \infty, \infty),$ and the vertical asymptote, $x = 0.$

Graphing a logarithmic function with the form f ( x ) = log _b ( x ).

Graph $f (x) = \log_{5} (x) .$ State the domain, range, and asymptote.

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

Since $b = 5$ is greater than one, we know the function is increasing. The left tail of the graph will approach the vertical asymptote $x = 0,$ and the right tail will increase slowly without bound.
The x -intercept is $(1, 0) .$
The key point $(5, 1)$ is on the graph.
We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points (see [link] ).

Graph of f(x)=log_5(x) with labeled points at (1, 0) and (5, 1). The y-axis is the asymptote.

The domain is $(0, \infty),$ the range is $(- \infty, \infty),$ and the vertical asymptote is $x = 0.$

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Try It

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

Graph of f(x)=log_(1/5)(x) with labeled points at (1/5, 1) and (1, 0). The y-axis is the asymptote.

The domain is $(0, \infty),$ the range is $(- \infty, \infty),$ and the vertical asymptote is $x = 0.$

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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 function $y = \log_{b} (x)$ without loss of shape.

Graphing a horizontal shift of f ( x ) = log _b ( x )

When a constant $c$ is added to the input of the parent function $f (x) = l o g_{b} (x),$ 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 $f (x) = \log_{b} (x)$ and for $c > 0$ alongside the shift left, $g (x) = \log_{b} (x + c),$ and the shift right, $h (x) = \log_{b} (x - c) .$ See [link] .

Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0 and g(x)=log_b(x+c) is the translation function with an asymptote at x=-c. This shows the translation of shifting left.

A General Note

Horizontal shifts of the parent function y = log _b ( x )

For any constant $c,$ the function $f (x) = \log_{b} (x + c)$

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

How To

Given a logarithmic function with the form $f (x) = \log_{b} (x + c),$ graph the translation.

Identify the horizontal shift:
1. If $c > 0,$ shift the graph of $f (x) = \log_{b} (x)$ left $c$ units.
2. If $c < 0,$ shift the graph of $f (x) = \log_{b} (x)$ right $c$ units.
Draw the vertical asymptote $x = - c .$
Identify three key points from the parent function. Find new coordinates for the shifted functions by subtracting $c$ from the $x$ coordinate.
Label the three points.
The Domain is $(- c, \infty),$ the range is $(- \infty, \infty),$ and the vertical asymptote is $x = - c .$

Graphing a horizontal shift of the parent function y = log _b ( x )

Sketch the horizontal shift $f (x) = \log_{3} (x - 2)$ alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.

Since the function is $f (x) = \log_{3} (x - 2),$ we notice $x + (- 2) = x - 2.$

Thus $c = - 2,$ so $c < 0.$ This means we will shift the function $f (x) = \log_{3} (x)$ right 2 units.

The vertical asymptote is $x = - (- 2)$ or $x = 2.$

Consider the three key points from the parent function, $(\frac{1}{3}, −1),$ $(1, 0),$ and $(3, 1) .$

The new coordinates are found by adding 2 to the $x$ coordinates.

Label the points $(\frac{7}{3}, −1),$ $(3, 0),$ and $(5, 1) .$

The domain is $(2, \infty),$ the range is $(- \infty, \infty),$ and the vertical asymptote is $x = 2.$

Graph of two functions. The parent function is y=log_3(x), with an asymptote at x=0 and labeled points at (1/3, -1), (1, 0), and (3, 1).The translation function f(x)=log_3(x-2) has an asymptote at x=2 and labeled points at (3, 0) and (5, 1).

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Questions & Answers

If c is the cost function for a particular product, find the marginal cost functions and their values at x=10 a. c(x) = 800+ 0.04x + 0.0002x² b. c(x) = 250 + 100x + 0.001x²

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how to reduce an equation?

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by manipulation of both side

9(y+8)-27 is 9y+45. Why can't you reduce that to y+5? I know that's wrong but can't explain why

Patrick Reply

when you reduce an equation to its simplest terms, you can't change the value of the equation. reducing it to y + 5 is equivalent to dividing it by 9 which changes the value. you can multiply it by 1 or 9/9 which would give 9(y + 5). multiplying it by one does not change the value.

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_3_2_1

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The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu

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1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3

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Q2 x+(x+2)+(x+4)=60 3x+6=60 3x+6-6=60-6 3x=54 3x/3=54/3 x=18 :. The numbers are 18,20 and 22

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Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?

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Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113

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Source: OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3

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6.4 Graphs of logarithmic functions (Page 3/8)

Graphing a logarithmic function with the form f ( x ) = log b ( x ).

Graphing transformations of logarithmic functions

Graphing a horizontal shift of f ( x ) = log b ( x )

Horizontal shifts of the parent function y = log b ( x )

Graphing a horizontal shift of the parent function y = log b ( x )

Questions & Answers

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Graphing a logarithmic function with the form f ( x ) = log _b ( x ).

Graphing a horizontal shift of f ( x ) = log _b ( x )

Horizontal shifts of the parent function y = log _b ( x )

Graphing a horizontal shift of the parent function y = log _b ( x )