We can use either the natural logarithm or the common logarithm to do this so we’ll do both.

\[{\log _{\frac{2}{3}}}53 = \frac{{\ln 53}}{{\ln \frac{2}{3}}} = \frac{{3.97029191}}{{ - 0.40546511}} = \require{bbox} \bbox[2pt,border:1px solid black]{{ - 9.79194469}}\] \[{\log _{\frac{2}{3}}}53 = \frac{{\log 53}}{{\log \frac{2}{3}}} = \frac{{1.72427587}}{{ - 0.17609126}} = \require{bbox} \bbox[2pt,border:1px solid black]{{ - 9.79194469}}\]

So, as we noted at the start it doesn’t matter which logarithm we use we’ll get the same answer in the end.