Use ${\log }_{36}6=0.5000$ , ${\log }_{36}4\approx 0.3869$ , and the properties of logarithms to approximate ${\log }_{36}144$ to four decimal places. Do not use a calculator.

A) $4.9678$
B) $4.5967$
C) $4.4506$
D) $3.4434$
E) $1.3869$

${\log }_{36}144$ can be simplified using the property of logarithms that ${\log }_b(a^n)=n\cdot {\log }_b(a)$ . Since $144=36^2$ , ${\log }_{36}144={\log }_{36}(36^2)=2\cdot {\log }_{36}36=2\cdot 1=2$ . However, none of the options match this directly. The given values can be used to approximate ${\log }_{36}144$ by expressing 144 as a product of 6 and 4, both of whose logarithms are given. Specifically, $144=6^2\cdot 4$ , so ${\log }_{36}144={\log }_{36}(6^2)+{\log }_{36}(4)=2\cdot {\log }_{36}(6)+{\log }_{36}(4)=2\cdot 0.5000+0.3869=1.3869$ .