y = 1/(1 + exp(–0.052(x – 95.63)))
What gets done last get undone first. That how you solve this.
Starting with x, one
first, subtracts 95.63 from it;
then, one multiplies by –0.052.;
then, one takes the exponential;
then, one adds 1;
then, one takes the reciprocal.
The result is y.
That gets you from x to y, and now the problem is to undo all of that going from y to x.
y = 1/(1 + exp(–0.052(x – 95.63)))
One undoes the last step, of taking the reciprocal, by taking the reciprocal again. This yields the following:
1/y = 1 + exp(–0.052(x – 95.63))
Then one undoes the addition of 1 by subtracting 1. This yields the following:
(1/y) – 1 = exp(–0.052(x – 95.63)).
Then one undoes the exponential by taking the logarithm. This yields the following:
log( (1/y) – 1) = –0.052(x – 95.62)
(By exp and log I mean respectively the base e exponential and base e logarithmic functions.)
Then one undoes the multiplication by –0.052 by dividing by –0.052. This yields the following:
( log( (1/y) – 1) ) / ( –0.052) = x – 95.62
or
x – 95.62 = ( 1 – log(1/y) ) / 0.052
or
x – 95.62 = ( 1 + log(y) ) / 0.052
(Since log(1/y) = –log(y).)
Then one undoes the subtraction of 95.62 by adding 95.62. This yields the following:
x = 95.62 + ( 1 + log(y) ) / 0.052.
One can write this as just one fraction by using 0.052 as the common denominator, and simplify via some arithmetic.
