SOLUTION: For the exponential function e^x and logarithmic function log x, graphically show the effect if x is doubled.

Algebra ->  Graphs -> SOLUTION: For the exponential function e^x and logarithmic function log x, graphically show the effect if x is doubled.       Log On


   

Question 156683: For the exponential function e^x and logarithmic function log x, graphically show the effect if x is doubled.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!


Jump to transformations of the logarithmic function

# 1

First let's calculate the values for y=e%5Ex


Let's find the y value when x=-2


y=e%5Ex Start with the given equation


y=e%5E%28-2%29 Plug in x=-2


y=0.135 Use a calculator to evaluate e%5E%28-2%29 to get 0.135


So if x=-2, then y=0.135. So we have the point (-2,0.135)


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Let's find the y value when x=-1


y=e%5Ex Start with the given equation


y=e%5E%28-1%29 Plug in x=-1


y=0.368 Use a calculator to evaluate e%5E%28-1%29 to get 0.368


So if x=-1, then y=0.368. So we have the point (-1,0.368)


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Let's find the y value when x=0


y=e%5Ex Start with the given equation


y=e%5E%280%29 Plug in x=0


y=1 Use a calculator to evaluate e%5E%280%29 to get 1


So if x=0, then y=1. So we have the point (0,1)


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Let's find the y value when x=1


y=e%5Ex Start with the given equation


y=e%5E%281%29 Plug in x=1


y=2.718 Use a calculator to evaluate e%5E%281%29 to get 2.718


So if x=1, then y=2.718. So we have the point (1,2.718)


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Let's find the y value when x=2


y=e%5Ex Start with the given equation


y=e%5E%282%29 Plug in x=2


y=7.389 Use a calculator to evaluate e%5E%282%29 to get 7.389


So if x=2, then y=7.389. So we have the point (2,7.389)



Now let's plot these points

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Now draw a line through the points to graph y=e%5Ex


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# 2

Now let's calculate the values for y=e%5E%282x%29


Let's find the y value when x=-2


y=e%5E%282x%29 Start with the given equation


y=e%5E%282%28-2%29%29 Plug in x=-2


y=e%5E%28-4%29 Multiply 2 and -2 to get -4


y=0.018 Use a calculator to evaluate e%5E%28-4%29 to get 0.018


So if x=-2, then y=0.018. So we have the point (-2,0.018)


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Let's find the y value when x=-1


y=e%5E%282x%29 Start with the given equation


y=e%5E%282%28-1%29%29 Plug in x=-1


y=e%5E%28-2%29 Multiply 2 and -1 to get -2


y=0.135 Use a calculator to evaluate e%5E%28-2%29 to get 0.135


So if x=-1, then y=0.135. So we have the point (-1,0.135)


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Let's find the y value when x=0


y=e%5E%282x%29 Start with the given equation


y=e%5E%282%280%29%29 Plug in x=0


y=e%5E%280%29 Multiply 2 and 0 to get 0


y=1 Use a calculator to evaluate e%5E%280%29 to get 1


So if x=0, then y=1. So we have the point (0,1)


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Let's find the y value when x=1


y=e%5E%282x%29 Start with the given equation


y=e%5E%282%281%29%29 Plug in x=1


y=e%5E%282%29 Multiply 2 and 1 to get 2


y=7.389 Use a calculator to evaluate e%5E%282%29 to get 7.389


So if x=1, then y=7.389. So we have the point (1,7.389)


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Let's find the y value when x=2


y=e%5E%282x%29 Start with the given equation


y=e%5E%282%282%29%29 Plug in x=2


y=e%5E%284%29 Multiply 2 and 2 to get 4


y=54.598 Use a calculator to evaluate e%5E%284%29 to get 54.598


So if x=2, then y=54.598. So we have the point (2,54.598)


Now let's plot these points

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Now draw a line through the points to graph y=e%5E%282x%29


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Notice how the graph of y=e%5E%282x%29 is much steeper than the graph of y=e%5E%28x%29. This is because the values of y=e%5E%282x%29 are simply the values of y=e%5E%28x%29 squared (since e%5E%282x%29=%28e%5E%28x%29%29%5E2). For instance, if x=1, then y=e%5E1=e and y=e%5E%282%2A1%29=e%5E2. Since e%5E2 is the square of e, this verifies our claim. So the transformation that occurs on y=e%5E%282x%29 is a horizontal compression (since we are dealing with "x" and the graph gets thinner).

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Jump to transformations of the exponential function
# 3


Now let's calculate the values for y=log%2810%2C%28x%29%29


Let's find the y value when x=1


y=log%2810%2C%28x%29%29 Start with the given equation


y=log%2810%2C%281%29%29 Plug in x=1


y=0 Use a calculator to evaluate log%2810%2C%281%29%29 to get 0


So if x=1, then y=0. So we have the point (1,0)

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Let's find the y value when x=2


y=log%2810%2C%28x%29%29 Start with the given equation


y=log%2810%2C%282%29%29 Plug in x=2


y=0.301 Use a calculator to evaluate log%2810%2C%282%29%29 to get 0.301


So if x=2, then y=0.301. So we have the point (2,0.301)

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Let's find the y value when x=3


y=log%2810%2C%28x%29%29 Start with the given equation


y=log%2810%2C%283%29%29 Plug in x=3


y=0.477 Use a calculator to evaluate log%2810%2C%283%29%29 to get 0.477


So if x=3, then y=0.477. So we have the point (3,0.477)

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Let's find the y value when x=4


y=log%2810%2C%28x%29%29 Start with the given equation


y=log%2810%2C%284%29%29 Plug in x=4


y=0.602 Use a calculator to evaluate log%2810%2C%284%29%29 to get 0.602


So if x=4, then y=0.602. So we have the point (4,0.602)

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Let's find the y value when x=5


y=log%2810%2C%28x%29%29 Start with the given equation


y=log%2810%2C%285%29%29 Plug in x=5


y=0.699 Use a calculator to evaluate log%2810%2C%285%29%29 to get 0.699


So if x=5, then y=0.699. So we have the point (5,0.699)



Now let's plot the points

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Now connect the points to graph y=log%2810%2C%28x%29%29
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# 4


Now let's calculate the values for y=log%2810%2C%282x%29%29


Let's find the y value when x=1


y=log%2810%2C%282x%29%29 Start with the given equation


y=log%2810%2C%282%2A1%29%29 Plug in x=1


y=log%2810%2C%282%29%29 Multiply 2 and 1 to get 2


y=0.301 Use a calculator to evaluate log%2810%2C%282%29%29 to get 0.301


So if x=1, then y=0.301. So we have the point (1,0.301)

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Let's find the y value when x=2


y=log%2810%2C%282x%29%29 Start with the given equation


y=log%2810%2C%282%2A2%29%29 Plug in x=2


y=log%2810%2C%284%29%29 Multiply 2 and 2 to get 4


y=0.602 Use a calculator to evaluate log%2810%2C%284%29%29 to get 0.602


So if x=2, then y=0.602. So we have the point (2,0.602)

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Let's find the y value when x=3


y=log%2810%2C%282x%29%29 Start with the given equation


y=log%2810%2C%282%2A3%29%29 Plug in x=3


y=log%2810%2C%286%29%29 Multiply 2 and 3 to get 6


y=0.778 Use a calculator to evaluate log%2810%2C%286%29%29 to get 0.778


So if x=3, then y=0.778. So we have the point (3,0.778)

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Let's find the y value when x=4


y=log%2810%2C%282x%29%29 Start with the given equation


y=log%2810%2C%282%2A4%29%29 Plug in x=4


y=log%2810%2C%288%29%29 Multiply 2 and 4 to get 8


y=0.903 Use a calculator to evaluate log%2810%2C%288%29%29 to get 0.903


So if x=4, then y=0.903. So we have the point (4,0.903)

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Let's find the y value when x=5


y=log%2810%2C%282x%29%29 Start with the given equation


y=log%2810%2C%282%2A5%29%29 Plug in x=5


y=log%2810%2C%2810%29%29 Multiply 2 and 5 to get 10


y=1 Use a calculator to evaluate log%2810%2C%2810%29%29 to get 1


So if x=5, then y=1. So we have the point (5,1)



Now let's plot the points

Photobucket - Video and Image Hosting


Now draw a curve through the points to graph y=log%2810%2C%282x%29%29

Photobucket - Video and Image Hosting




By comparing y=log%2810%2C%28x%29%29 and y=log%2810%2C%282x%29%29, we can see that the graph of y=log%2810%2C%282x%29%29 is simply the graph of y=log%2810%2C%28x%29%29 shifted up some number of units. So a vertical translation has occured on y=log%2810%2C%28x%29%29 to get y=log%2810%2C%282x%29%29