# Thread: Help rearranging complex formula

1. ## 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 = ......

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

2. ## Re: Help rearranging complex formula

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

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

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

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

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

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

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

3. ## 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

4. ## Re: Help rearranging complex formula

That domain after transformation has shrunk is not strange .Expotetial is not surjection

5. ## 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:

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:

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?

6. ## Re: Help rearranging complex formula

Originally Posted by sshinkwin
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:

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:

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.