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Math Help - Help rearranging complex formula

  1. #1
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    Help rearranging complex formula

    Hi all,

    I have an issue trying to rearrange this formula.

    X = ( Z / (1 + 81 ^ ( ( A + B/2 - Y ) / B ) ) )

    I am trying to rearrange this formula to make A the subject i.e. A = ......

    I can rearrange this formula to start with to get:

    Z / X - 1 = 81 ^ ((A + (B/2) - Y) / B)

    I also know that X = Y ^Z and that Y = X (1/Z). But i believe in this case you need to you LOG but i would struggle with this next step.

    Your help here would be appreciated.

    Thanks

    Stewart
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  2. #2
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    Re: Help rearranging complex formula

    $X=\dfrac{Z}{81^{\frac{A+B/2-Y}{B}}+1}$

    $1+81^{\frac{A+B/2-Y}{B}}=\dfrac{Z}{X}$

    $81^{\frac{A+B/2-Y}{B}}=\dfrac{Z}{X}-1$

    $3^{4\frac{A+B/2-Y}{B}}=\dfrac{Z}{X}-1$

    $4\dfrac{A+B/2-Y}{B}=\log_3\left(\dfrac{Z}{X}-1\right)$

    $A+B/2 - Y=\dfrac{B}{4}\log_3\left(\dfrac{Z}{X}-1\right)$

    $A=\dfrac{B}{4}\log_3\left(\dfrac{Z}{X}-1\right)-B/2+Y$
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  3. #3
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    Re: Help rearranging complex formula

    Thanks, that works.

    The original equation i had an inverse to that as well which was:

    X = ( - Z ) / ( 1 + 81 ^ P - Z ) , where P = [ A + ( B / 2 ) - Y ] / B

    I subsequently rearrange to 81 ^ P = ( Z / - X ) + Z - 1

    If A ^ X = B can be re-arranged to X = LOG (B) / LOG (A) then,

    P = LOG [ ( Z / -X ) + Z - 1 ] / LOG [ 81 ]

    However this is not working because ( Z / -X ) + Z - 1 is a negative value.

    Does anyone have any suggestions how to solve this. I am trying to make A the subject.

    Thanks
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  4. #4
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    Re: Help rearranging complex formula

    That domain after transformation has shrunk is not strange .Expotetial is not surjection
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  5. #5
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    Re: Help rearranging complex formula

    I had my origingal formula: X = ( Z / (1 + 81 ^ ( ( A + B / 2 - Y ) / B ) ) ) . I did not create this, this was taken from the internet.

    Where:
    A = Start of growth from 10% of saturation level
    B = Time to get from 10% to 90% of saturation level
    X = Adoption % in given year
    Y = Given year
    Z = Saturation level

    With values of A = 2016.5 , B = 4 , Z = 100% it produce this adoption curve:

    Help rearranging complex formula-adoption-curve.png

    I was then able to rearrange this formula to show A as the subject: A = B * LOG [ ( Z - X ) / X ] / LOG ( 81 ) - ( B / 2 ) + Y

    I then adjusted this original formula to show decay: X = ( - Z / (1 + 81 ^ ( ( A + B / 2 - Y ) / B ) ) ) – Z . With the same values as before this gave me a curve like:

    Help rearranging complex formula-decay-curve.png

    I now want to re-arrange this decay formula to have A as the subject.

    This is where I am struggling as I end up with the LOG of a negative value. Or have I rearranged incorrectly? Is there another way to show decay given my original equation?
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  6. #6
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    Re: Help rearranging complex formula

    Quote Originally Posted by sshinkwin View Post
    I had my origingal formula: X = ( Z / (1 + 81 ^ ( ( A + B / 2 - Y ) / B ) ) ) . I did not create this, this was taken from the internet.

    Where:
    A = Start of growth from 10% of saturation level
    B = Time to get from 10% to 90% of saturation level
    X = Adoption % in given year
    Y = Given year
    Z = Saturation level

    With values of A = 2016.5 , B = 4 , Z = 100% it produce this adoption curve:

    Click image for larger version. 

Name:	Adoption Curve.png 
Views:	1 
Size:	6.3 KB 
ID:	31462

    I was then able to rearrange this formula to show A as the subject: A = B * LOG [ ( Z - X ) / X ] / LOG ( 81 ) - ( B / 2 ) + Y

    I then adjusted this original formula to show decay: X = ( - Z / (1 + 81 ^ ( ( A + B / 2 - Y ) / B ) ) ) Z . With the same values as before this gave me a curve like:

    Click image for larger version. 

Name:	Decay Curve.png 
Views:	0 
Size:	7.9 KB 
ID:	31463

    I now want to re-arrange this decay formula to have A as the subject.

    This is where I am struggling as I end up with the LOG of a negative value. Or have I rearranged incorrectly? Is there another way to show decay given my original equation?
    $\forall \alpha \in \mathbb{R}~~81^\alpha > 0 \Rightarrow \dfrac X Z -1> 0$

    the log of a negative value shouldn't be required.
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