document.write( "Question 174032: Can someone help me graph this please? I have the hardest time with graphing. I can usually figure the problem out, but the graphing is where I struggle. Thank you so much.
\n" ); document.write( "Exponential function: Logarithmic function:
\n" ); document.write( "f(x)=y=b^x f^-1*(x)=y=logbx
\n" ); document.write( "now if b=4 the vertical line is y=4^x and the horizontal line is f(x)=log4x \n" ); document.write( "
Algebra.Com's Answer #129099 by gonzo(654)\"\" \"About 
You can put this solution on YOUR website!
\"y+=+b%5Ex\"
\n" ); document.write( "if and only if:
\n" ); document.write( "\"x+=+log%28b%2Cy%29\"
\n" ); document.write( "substituting x for y and y for x, the inverse function is therefore:
\n" ); document.write( "\"y+=+log%28b%2Cx%29\"
\n" ); document.write( "-----
\n" ); document.write( "as you state, if b = 4, then:
\n" ); document.write( "exponential equation is:
\n" ); document.write( "\"y+=+4%5Ex\"
\n" ); document.write( "logarithmic equation is:
\n" ); document.write( "\"y+=+log%284%2Cx%29\"
\n" ); document.write( "-----
\n" ); document.write( "since both of these equations are in graphable form, if you have a graphing calculator, you just plug in the formulas:
\n" ); document.write( "4^x
\n" ); document.write( "and
\n" ); document.write( "log(4,x)
\n" ); document.write( "and the graphing caculator will take care of if for you.
\n" ); document.write( "you just have to make sure you input the formulas correctly for the algorithm that is translating the equation for you.
\n" ); document.write( "-----
\n" ); document.write( "i'll show you the graph and then give you some words below it.
\n" ); document.write( "the graph is:
\n" ); document.write( "\"graph%28600%2C600%2C-10%2C10%2C-10%2C10%2C4%5Ex%2Clog%284%2Cx%29%29\"
\n" ); document.write( "since these are inverse functions, the solution (a,b) for one equation should be equal to (b,a) for the other equation.
\n" ); document.write( "example:
\n" ); document.write( "let x = 5
\n" ); document.write( "then y = \"4%5E5\" = 1024
\n" ); document.write( "-----
\n" ); document.write( "for the inverse equation,
\n" ); document.write( "let x = 1024
\n" ); document.write( "then y =\"log%284%2C1024%29\" = \"log%2810%2C1024%29%2Flog%2810%2C4%29\" = 5
\n" ); document.write( "-----
\n" ); document.write( "your first solution set is: (5,1024)
\n" ); document.write( "your inverse solution set is: (1024,5)
\n" ); document.write( "-----
\n" ); document.write( "this proves these functions are inverse to each other.
\n" ); document.write( "-----
\n" ); document.write( "to graph manually, since they are both in y = f(x) form, you simply take values for x and solve in each equation:
\n" ); document.write( "then you plot the graph.
\n" ); document.write( "-----
\n" ); document.write( "with your exponential equation, x can be anything.
\n" ); document.write( "with your logarithmic equation, x has to be greater than 0.
\n" ); document.write( "plot every 5 values of x to see where the equations are going:
\n" ); document.write( "first take x = 0:
\n" ); document.write( "\"4%5E0\" = 1:
\n" ); document.write( "\"log%284%2C0%29\" is not possible.
\n" ); document.write( "then take x = 5:
\n" ); document.write( "\"4%5E5\" = 1024
\n" ); document.write( "\"log%284%2C5%29\" = 1.16...
\n" ); document.write( "take x = 10:
\n" ); document.write( "\"4%5E10\" = 1048576 which is off the chart (way up there)
\n" ); document.write( "\"log%284%2C10%29\" = 1.66...
\n" ); document.write( "this is going up much less dramatically.
\n" ); document.write( "---
\n" ); document.write( "you can take some x points in between to fill it out and get a more accurate representation of what the curve will look like.
\n" ); document.write( "if you go minus,
\n" ); document.write( "then take x = -5
\n" ); document.write( "\"4%5E-5\" = .000976...
\n" ); document.write( "\"log%284%2C-5%29\" is not possible.
\n" ); document.write( "-----
\n" ); document.write( "what this should be telling you is that as x gets more negative, the exponential equation approaches 0 and the logarithmic equation is not there.
\n" ); document.write( "-----
\n" ); document.write( "you can see all of this from the graph that i just displayed for you above.
\n" ); document.write( "-----
\n" ); document.write( "if you don't have a graphing calculator, you can use an online graphing calculator like the one at the following website address:
\n" ); document.write( "http://www.e-tutor.com/et3/graphing
\n" ); document.write( "this particular calculator can't model log to bases other than 10, but you can use the conversion formula to convert to the base 10.
\n" ); document.write( "that conversion formula is:
\n" ); document.write( "\"log%28b%2C%28x%29%29+=+log%28%28x%29%29%2Flog%28%28b%29%29\"
\n" ); document.write( "you separate equations using semi:colon.
\n" ); document.write( "for example:
\n" ); document.write( "i went to that website and entered the following into the formula section:
\n" ); document.write( "4^x;log(x)/log(4)
\n" ); document.write( "try it yourself if you have time and interest.
\n" ); document.write( "-----
\n" ); document.write( "if you wish to see how the formula for the graph was modeled in algebra.com, just look at the source code and look for the keyword \"graph\".
\n" ); document.write( "----- \n" ); document.write( "
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