SOLUTION: I would like to get some assistance with this question. Give an example of an exponential function. Convert this exponential function to a logarithmic function. Plot the graph. t

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: I would like to get some assistance with this question. Give an example of an exponential function. Convert this exponential function to a logarithmic function. Plot the graph. t      Log On


   

Question 172962: I would like to get some assistance with this question. Give an example of an exponential function. Convert this exponential function to a logarithmic function. Plot the graph.
thank you
Found 2 solutions by solver91311, gonzo:
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!
f%28x%29=y=b%5Ex is an exponential function. Notice that the independent variable is an exponent.

The inverse function is f%5E-1%28x%29=y=log%28b%2Cx%29

Selecting b = 10, the red line is the graph of y=10%5Ex and the green line is the graph of y=log%2810%2Cx%29

graph%28600%2C600%2C-5%2C5%2C-5%2C5%2C10%5Ex%2Clog%2810%2Cx%29%29

Note that there are imperfections in the rendering of these graphs. The exponential function is asymptotic to the negative x-axis, and the log function is asymptotic to the negative y-axis.









Answer by gonzo(654) About Me  (Show Source):
You can put this solution on YOUR website!
the general form of an exponential function is:
y+=+b%5Ex
where:
b is the base
and
x is the exponent
and
y is the result of the base raised to the exponent.
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the general form of a logarithmic function is:
y+=+log%28b%2Cx%29
where:
b is the base
and
y is the exponent
and
x is the result of the base raised to the exponent.
-----
the rule for logarithms is:
y+=+b%5Ex if and only if: x+=+log%28b%2Cy%29
conversely:
y+=+log%28b%2Cx%29 if and only if: x+=+b%5Ey%29
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the exponential form of the equation and the logarithmic form of the equation are inverses of each other.
here's an example:
10^2 = 100
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the exponential form of this equation is:
y+=+10%5Ex
x = 2
y = 100
coordinate point on the graph of this equation is: (2,100)
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the logarithmic form of this equation is:
y+=+log%2810%2Cx%29
x = 100
y = 2
coordinate point on the graph of this equation is: (100,2)
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in general:
if f(x) = (a,b), and g(x) = (b,a) then the equations are inverses of each other.
this is true for the two equations:
y+=+f%28x%29+=+b%5Ex
and
y+=+g%28x%29+=+log%28b%2Cx%29
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you can also graph these functions (i used the base of 10 - any base would work).
the functions are:
y+=+10%5Ex
and
y+=+log%2810%2Cx%29
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we used base b = 10 so that it can be graphed easily and solved for easily on the calculator. the base can be any real number > 0.
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the line y = x is inserted in the middle of the graph to show more clearly that these functions are inverses of each other which shows up as their being reflections about the line y = x (kind of like mirror images if you think of the line y = x as the mirror).
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look below the graph for further comments.
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graph%28800%2C800%2C-10%2C10%2C-10%2C10%2C10%5Ex%2Clog%2810%2Cx%29%2Cx%29
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a practical example:
suppose you are given:
4500+=+10%5Ex
how would you solve this?
let y = 4500
let x = x
let b (the base) = 10
using the laws of logarithms, you know that:
4500+=+10%5Ex
if and only if:
x+=+log%2810%2C4500%29
so you go to the calculator and get the log of 4500 which is: 3.653212514
how do you know this is good?
you use the calculator again to take 10%5E3.653212514 to get: 4500
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what if the base is not 10?
how to solve?
there's a conversion formula you can use to convert any base to the base of 10 so you can solve it on your calculator.
that conversion formula is:
log%28b%2Cx%29+=+log%2810%2Cx%29%2Flog%2810%2Cb%29
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here's an example of how that works.
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your new problem:
34561+=+3.5%5Ex
you know that this is true if and only if:
log%283.5%2C34561%29+=+x
you can't solve that on your calculator directly, but you can convert it to the base of 10 using the conversion formula.
that would be:
log%283.5%2C34561%29+=+log%2810%2C34561%29%2Flog%2810%2C3.5%29+=+x
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you use your calculator to get:

which becomes:
x = 8.341946099
this means that:
34561+=+3.5%5E8.341946099
you can use your calculator to prove that this is true:
34561 = 34561 proving that it is true.
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