How are logarithmic trendline equations used to determine antibody concentrations in ELISA?
If by ln you mean the natural logarithm, then no, you can't rearrange the equation in that way. Try this:

$$m\cdot \ln(x) - c = y\\
\ln(x) = y + c\\
\ln(x) = \frac{y+c}{m}\\
\exp(\ln(x)) = \exp\left(\frac{y+c}{m}\right)\\
x = \exp\left(\frac{y+c}{m}\right)
$$