$z = \frac{xy\ln(a)\ln(b)}{\ln(a)+\ln(b)}$

$z = \frac{x\ln(a) + y\ln(b)}{\ln(c)}$

$z = \frac{xy(\ln(a)+\ln(b))}{x\ln(b)+y\ln(a)}$

$z = \frac{x\ln(a)}{y\ln(b)}$