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

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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)   (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


Let's find the y value when


Start with the given equation


Plug in


Use a calculator to evaluate to get


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


--------------------------------


Let's find the y value when


Start with the given equation


Plug in


Use a calculator to evaluate to get


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


--------------------------------


Let's find the y value when


Start with the given equation


Plug in


Use a calculator to evaluate to get


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


--------------------------------


Let's find the y value when


Start with the given equation


Plug in


Use a calculator to evaluate to get


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


--------------------------------


Let's find the y value when


Start with the given equation


Plug in


Use a calculator to evaluate to get


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



Now let's plot these points

Photobucket - Video and Image Hosting


Now draw a line through the points to graph


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# 2

Now let's calculate the values for


Let's find the y value when


Start with the given equation


Plug in


Multiply and to get


Use a calculator to evaluate to get


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


--------------------------------


Let's find the y value when


Start with the given equation


Plug in


Multiply and to get


Use a calculator to evaluate to get


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


--------------------------------


Let's find the y value when


Start with the given equation


Plug in


Multiply and to get


Use a calculator to evaluate to get


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


--------------------------------


Let's find the y value when


Start with the given equation


Plug in


Multiply and to get


Use a calculator to evaluate to get


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


--------------------------------


Let's find the y value when


Start with the given equation


Plug in


Multiply and to get


Use a calculator to evaluate to get


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


Now let's plot these points

Photobucket - Video and Image Hosting


Now draw a line through the points to graph


Photobucket - Video and Image Hosting


Notice how the graph of is much steeper than the graph of . This is because the values of are simply the values of squared (since ). For instance, if , then and . Since is the square of , this verifies our claim. So the transformation that occurs on is a horizontal compression (since we are dealing with "x" and the graph gets thinner).

======================================================



Jump to transformations of the exponential function
# 3


Now let's calculate the values for


Let's find the y value when


Start with the given equation


Plug in


Use a calculator to evaluate to get


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

----------------------------------------

Let's find the y value when


Start with the given equation


Plug in


Use a calculator to evaluate to get


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

----------------------------------------

Let's find the y value when


Start with the given equation


Plug in


Use a calculator to evaluate to get


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

----------------------------------------

Let's find the y value when


Start with the given equation


Plug in


Use a calculator to evaluate to get


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

----------------------------------------

Let's find the y value when


Start with the given equation


Plug in


Use a calculator to evaluate to get


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



Now let's plot the points

Photobucket - Video and Image Hosting


Now connect the points to graph
Photobucket - Video and Image Hosting


=============================================================================

# 4


Now let's calculate the values for


Let's find the y value when


Start with the given equation


Plug in


Multiply and to get


Use a calculator to evaluate to get


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

----------------------------------------

Let's find the y value when


Start with the given equation


Plug in


Multiply and to get


Use a calculator to evaluate to get


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

----------------------------------------

Let's find the y value when


Start with the given equation


Plug in


Multiply and to get


Use a calculator to evaluate to get


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

----------------------------------------

Let's find the y value when


Start with the given equation


Plug in


Multiply and to get


Use a calculator to evaluate to get


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

----------------------------------------

Let's find the y value when


Start with the given equation


Plug in


Multiply and to get


Use a calculator to evaluate to get


So if , then . 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

Photobucket - Video and Image Hosting




By comparing and , we can see that the graph of is simply the graph of shifted up some number of units. So a vertical translation has occured on to get
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