Request for the equation about log scale

[wjd582]

Hi
I would like to know the equation to find x when given y.

the data is like below.

x y
0.5 5000
1.5 5020.3
2.5 5040.683
3.5 5061.148
4.5 5081.696
'' ''
'' ''
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2048.5 20000000

the interval of y is log scale from 5000 to 2 x 10^7.

My question is how to know the value of X when I know the value Y.

For example, when the Y is given 1000000, what is the X?

Please help me to solve this problem.

[Guest]

You seem to be saying that y if an exponential function of x. That is y= C(10^{Ax}) for some values of A and C.
We have three unknowns, A and C, so we need two equations. To get as representative a sample as possible, I will take them from x= 0.5, y= 5000 and x= 2048.5, y= 20000000.

So [tex]5000= C(10^{0.5A}[/tex] and [tex]20000000= C(10^{2048.5A})[/tex]

Dividing the second equation by the first eliminates C: [tex]4000= 10^{2047.92A}[/tex].

Taking the logarithm, base 10, 3.60206= 2047.92A so A= 3.60206/2047.92= 0.00175889.

Then [tex]5000= C(10^{0.5A}= C(10^{0.000879})= 1.00203C[/tex] so C= 5000/1.00203= 4990.

The function is, at least approximately, [tex]y= 4990(10^{0.00175889x})[/tex] so that= if y= 1000000 then
[tex]1000000= 4990(10^{0.00175880x})[/tex]. Divide both sides by 4990: [tex]\frac{1000000}{4990}= 200.4= 10^{0.00175880x}[/tex]. Take the logarithm, base 10, of both sides: [tex]lof(200.4)= 2.3019= 0.00175889x[/tex]. Finally, divide both sides by 0.0017589; [tex]x= \frac{2.3019}{0.00175889}= 1309[/tex].

When y is 1000000, x is 1309.