Ta có: ${\log }_{a}b=2{\log }_{b}c\iff {\log }_{a}b\cdot {\log }_{b}c=2{\log }_b^{2}c\iff {\log }_{a}c=2{\log }_b^{2}c$

Ta có: ${\log }_{a}b=4{\log }_{c}a\iff {\log }_{a}b\cdot {\log }_{c}a=4{\log }_c^{2}a\iff {\log }_{c}b=4{\log }_c^{2}a$.

Suy ra ${\log }_c$b. ${\log }_{a}c=8{\log }_c^{2}a\cdot {\log }_b^{2}c\iff {\log }_{a}b=8{\log }_b^{2}a$

Mặt khác: ${\log }_{a}b=2{\log }_{b}c\iff {\log }_a{a}^2=2{\log }_{b}c\iff {\log }_{b}c=1\iff b=c$.

Theo già thiết: $a+2b+3c=48\iff a+2a^2+3a^2=48\iff 5a^2+a-48=0\iff \left[\begin{array}{l}a=3\\ a=-\frac{16}{5}\end{array}\right.$.