Question 212950
y = 2^x x = 2^y
Evaluate the exponential equation for three positive values of x, three negative values of x, and x =0. Transform the second expression into the equivalent logarithmic equation; and evaluate the logarithmic equation for three values of x that ar greater than 1, three values of x that are between 0 and 1, and at x = 1. Use the resulting ordered pairs to plot the graph of each function.
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FIRST EQUATION
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{{{y = 2^x}}}
x is the exponent, 2 is the base.
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when x is positive:
{{{2^1 = 2}}}
{{{2^2 = 4}}}
{{{2^3 = 8}}}
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when x is negative:
{{{2^(-1) = 1/(2^1) = 1/2}}}
{{{2^(-2) = 1/(2^2) = 1/4}}}
{{{2^(-3) = 1/(2^3) = 1/8}}}
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graph of y = 2^x is shown below:
{{{graph (300,300,-10,10,-10,10,2^x)}}}
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SECOND EQUATION
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{{{x = 2^y}}}
y is the exponent, 2 is the base.
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when y is positive:
{{{2^1 = 2}}}
{{{2^2 = 4}}}
{{{2^3 = 8}}}
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when y is negative:
{{{2^(-1) = 1/(2^1) = 1/2}}}
{{{2^(-2) = 1/(2^2) = 1/4}}}
{{{2^(-3) = 1/(2^3) = 1/8}}}
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You cannot graph this directly since the graphs are set up so that the independent variable is x and the dependent variable is y.
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In order to graph this, you have to solve for y.
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Your equation is:
{{{x = 2^y}}}
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The basic definition of an exponential equation is:
{{{y = b^x}}} if and only if {{{x = log(b,y)}}}
When you reverse the x and y like we have in this second form of the exponential equation, then the basic definition becomes:
{{{x = b^y}}} if and only if {{{y = log(b,x)}}}
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Your equation is:
{{{x = 2^y}}}
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By the basic definition of exponents, {{{x = 2^y}}} if and only if {{{y = log(2,x)}}}
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The graph of {{{y = log(2,x)}}} is shown below:
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{{{graph (300,300,-10,10,-10,10,log(2,x))}}}
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{{{y = log(2,x)}}} is the inverse equation of {{{y = 2^x}}}
You derive the inverse equation by doing exactly what they are asking you to do when they showed you the two equations.
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In order to find the inverse equation of {{{y = 2^x}}} you would do the following:
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Solve for x:
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By the basic definition of exponents, {{{y = 2^x}}} if and only if {{{x = log(2,y)}}}
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You have just solved for x by just applying the basic definition of exponents.
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You have:
{{{x = log(2,y)}}}
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Next you transpose the x and the y in the equation.
Your equation becomes:
{{{y = log(2,x)}}}
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{{{y = log(2,x)}}} is the inverse equation of {{{y = 2^x}}}
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You have just derived the inverse equation of {{{y = 2^x}}}.  That equation is {{{y = log(2,x)}}}
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The inverse equation is the reflection of the normal equation about the line y = x.
The following graph shows the normal equation of {{{y = 2^x}}} and the inverse equation of {{{y = log(2,x)}}} and the line {{{y = x}}}
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{{{graph (300,300,-10,10,-10,10,2^x,log(2,x),x)}}}
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If you draw a perpendicular line anywhere through the line y = x, the intersection of that line through {{{y = 2^x}}} will be (a,b), and the intersection of that line through {{{y = log(2,x)}}} will be (b,a).
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for example:
when x = 2, {{{y = 2^2}}} = 4 so the coordinates are (2,4)
when x = 4, {{{y = log(2,4)}}} = 2 so the coordinates are (4,2)
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The equation of the line perpendicular to the line y = x and passing through the points (2,4) and (4,2) is y = -x + 6.
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Adding that line to the graph of the 3 equations yields the following:
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{{{graph (300,300,-10,10,-10,10,2^x,log(2,x),x,-x+6)}}}
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As you can see, the two points intersecting with the line perpendicular to the line y = x are the same distance from that line making the two graphs symmetric to each other about the line y = x.
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