\begin{align} \chi^2 &= \sum_i \left[ \ln(y_i-c) - \ln a +bx_i \right]^2\\ &= \sum_i \left[ \left[ \ln(y_i-c) \right]^2 + \left(\ln a\right)^2 + b^2 x_i^2 - 2 \ln a\ln(y_i-c) + 2 bx_i\ln(y_i-c) - 2b x_i \ln a\right]\\ \end{align}

\begin{align} \sum_i \left[ \ln a - \ln(y_i-c) - b x_i \right] = 0\\ \sum_i \left[ b x_i^2 + x_i \ln\left( y_i-c\right) - x_i \ln a \right] = 0\\ \sum_i \left[ \frac{\ln(y_i-c)}{y_i-c} - \frac{\ln a}{y_i-c} + \frac{b x_i}{y_i-c}\right] = 0\\ \end{align}

\begin{align} n\ln a - \sum_i \ln(y_i-c) - b \sum_i x_i = 0\\ b \sum_i x_i^2 + \sum_i x_i \ln\left( y_i-c\right) - \ln a \sum_i x_i = 0\\ \sum_i \frac{\ln(y_i-c)}{y_i-c} - \ln a \sum_i \frac{1}{y_i-c} + b \sum_i \frac{x_i}{y_i-c}= 0\\ \end{align}

\begin{align} n \ln a \sum_i x_i - \sum_i x_i \sum_i \ln(y_i-c) - b \left(\sum_i x_i\right)^2 = 0\\ n b \sum_i x_i^2 + n\sum_i x_i \ln\left( y_i-c\right) - n\ln a \sum_i x_i = 0\\ \end{align}