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Logarithm Function Reverse Mode Theory
We use the reverse theory standard math function definition for the functions  H and  G . The forward mode formulas for the logarithm function are  \[
     z^{(j)}  =  \log ( x^{(0)} ) 
\] 
for the case  j = 0 , and for  j > 0 ,  \[
z^{(j)} 
=  \frac{1}{ x^{(0)} } \frac{1}{j} 
\left(
     j x^{(j)}
     - \sum_{k=1}^{j-1} k z^{(k)} x^{(j-k)}  
\right)
\] 
otherwise. If  j = 0 , we have the relation  \[
\D{H}{ x^{(j)} } = 
\D{G}{ x^{(j)} }  + \D{G}{ z^{(j)} } \frac{1}{ x^{(0)} }
\] 
If  j > 0 , then for  k = 1 , \ldots , j-1  \[
\begin{array}{rcl}
\D{H}{ x^{(0)} } & = &
\D{G}{ x^{(0)} } - \D{G}{ z^{(j)} } \frac{1}{ x^{(0)} } 
\frac{1}{ x^{(0)} } \frac{1}{j} 
\left(
     j x^{(j)}
     - \sum_{m=1}^{j-1} m z^{(m)} x^{(j-m)}  
\right)
\\
& = &
\D{G}{ x^{(0)} } - \D{G}{ z^{(j)} } \frac{1}{ x^{(0)} } z^{(j)}
\\
\D{H}{ x^{(j)} } & = & 
\D{G}{ x^{(j)} } + \D{G}{ z^{(j)} } \frac{1}{ x^{(0)} } 
\\
\D{H}{ x^{(j-k)} } & = & 
\D{G}{ x^{(j-k)} }  - 
     \D{G}{ z^{(j)} } \frac{1}{ x^{(0)} } \frac{1}{j}  k z^{(k)}
\\
\D{H}{ z^{(k)} } & = & 
\D{G}{ z^{(k)} }  - 
     \D{G}{ z^{(j)} } \frac{1}{ x^{(0)} } \frac{1}{j}  k x^{(j-k)}
\end{array}
\] 

Input File: omh/log_reverse.omh