Answer
$z\approx 4.183$
Work Step by Step
$3^{z-2}=17\qquad \color{blue}{\text{ ...apply ln(...) to both sides } }$
$\ln 3^{z-2}=\ln 11\qquad \color{blue}{\text{ ...apply } \log_{a}x^{r}=r\log_{a}x}$
$(z-2)\ln 3=\ln 11\qquad \color{blue}{\text{ ... } /\div\ln 3}$
$z-2=\displaystyle \frac{\ln 11}{\ln 3}$
$z=\displaystyle \frac{\ln 11}{\ln 3}+2$
$z\approx 4.183$