Question

If ${\log }_{6}16=k,then{\log }_{18}24$ =

Answer 1

${\log }_{6}16=k$

${\log }_6(2{)}^4=k$

$4{\log }_{6}2=k$

${\log }_{6}2=\frac{k}{4}$

Use change of base formula to find ${\log }_{18}24$

${\log }_{b}a=\frac{{\log }_{x}a}{{\log }_{x}b}$

${\log }_{18}24=\frac{{\log }_{6}24}{{\log }_{6}18}$

$=\frac{{\log }_6(6\times 4)}{{\log }_6\left(\frac{36}{2}\right)}$

$=\frac{{\log }_{6}6+{\log }_{6}4}{{\log }_{6}36-{\log }_{6}2}$

$=\frac{{\log }_{6}6+{\log }_6(2{)}^2}{{\log }_6(6{)}^2-{\log }_{6}2}$

$=\frac{{\log }_{6}6+2{\log }_{6}2}{2{\log }_{6}6-{\log }_{6}2}$

$=\frac{1+\frac{2k}{4}}{2-\frac{k}{4}}$

$=\frac{4+2k}{8-k}$