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?