5-Minute Check on Activity 5-10

1
5-Minute Check on Activity 5-10

• Convert the following from or to logarithmic
  equations
• y 10x
• log10 x 2.16
• Using your calculator find the following to two
  decimal places
• log 34
• log 25
• ln 32
• ln 7
• ln 17

log10 y x
102.16 x
1.53
1.40
3.47
1.95
2.83
Click the mouse button or press the Space Bar to
display the answers.
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Activity 5 -11
Bay Street, Toronto, Canada 5/27/2005

• Walking Speed of Pedestrians

3
Objectives

• Determine the inverse of the exponential function
• Identify the properties of the graph of a
  logarithmic function
• Graph the natural logarithmic function using
  transformations

4
Vocabulary

• Logarithmic function defined by the inverse
  function of an exponential function to the base,
  b
• Inverse functions interchange the domain and
  range of the original function. The graph of an
  inverse function is the reflection of the
  original function about the line y x. You
  determine the inverse function by switching x and
  y and solve the new equation for y.
• Vertical asymptote a vertical line that the
  function approaches but never reaches (usually a
  restricted value in the domain)

5
Activity

• On a recent visit to Boston, you notice that
  people seem rushed as the move about the city.
  Upon returning to school, you mention this
  observation to your psychology teacher. The
  instructor refers you to a psychology study that
  investigates the relationship between the average
  walking speed of pedestrians and the population
  of the city.
• Does the data appear to be linear?
• Does the data appear to be exponential?

No rate of change is not constant
No rate of change is not increasing as x
increases
6
Activity cont

• The study cites statistics presented in the graph.

Data is logarithmic
7
Common Logarithmic Function

• The logarithmic function base b is defined by
• y logb x
• where b is the base (b gt 0 and b ? 1)
• x is the power of the base, b and
• y is the exponent needed on b to obtain x
• Properties of f(x) log x
  (common log)
• Base is 10
• Domain of x is set of all positive numbers (x gt
  0)
• Range of y is set of all real numbers
• f(x) is inverse of g(x) 10x

8
Natural Logarithmic Functions

• Properties of f(x) ln x
• Base is e (natural number 2.718281828)
• Domain of x is set of all positive numbers (x gt
  0)
• Range of y is set of all real numbers
• f(x) is inverse of g(x) ex

9
Natural Logarithmic Graph

• Fill in the table and graph y ln x

x 0.01 0.1 0.25 0.5 1 5 10 20 50
y ln x
-4.6 -2.3 -1.4 -0.7 0 1.61 2.30 2.99 3.91
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Inverse Functions

• Inverse functions interchange the domain and
  range of the original function. Also the graph
  of the inverse function is a reflection of the
  original function about the line y x.
• To get the inverse function, simply switch the x
  and y values in the equation and solve the new
  equation for y.
• Fill in the table below using your calculator

x f(x) 10x x g(x) log x
-2 0.01
-1 0.1
0 1
1 10
2 100
0.01 0.1 1 10 100
-2 -1 0 1 2
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Inverse Functions Examples

• Find the inverse of the following functions
• y 2x 3
• y x2
• y 5x

x 2y 3 x 3 2y ½ x
3/2 y
x y2 vx y
x 5y Y log5 x
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Vertical Asymptotes

• A vertical asymptote is a vertical line, x a,
  that the graph of a function becomes very close
  to but never touches. As the x-values get closer
  and closer to a, the y-values get larger and
  larger in magnitude (approaching positive or
  negative infinity, 8).
• These are usually at restricted values (division
  by zero or negatives under the radical) in the
  domain.

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Logarithmic Transformations

• Compared to y ln x, describe the graphic
  relationship between its graph and the following
  graphs
• y -ln x
• y ln (x 3)
• y ln (-x)
• y ln x 3
• y 2 ln x
• y 3 2 ln x

Outside Reflection across x-axis
Inside Shift left 3 units
Inside Reflection across y-axis
Outside Shift up 3 units
Outside Vertical stretch by 2
Outside Vertical stretch by 2 reflected across
x-axis and shifted up by 3
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Summary and Homework

• Summary
• Properties of the function y log x
• Domain is x gt 0
• Range is all real numbers
• Is the inverse of the function x 10y
• Graph of the function y logb x or y ln x
• is increasing for all x gt 0
• x-intercept of (1,0) and no y-intercept
• x 0 (y-axis) is a vertical asymptote
• is continuous for x gt 0
• Inverse functions are determined by interchanging
  x and y variables
• Homework
• pg 640-43 problems 1, 3-5