Question Number 105269 by malwaan last updated on 27/Jul/20
$${a};{b};{c}\:{are}\:{real}\:{numbers} \\ $$ $$\mathrm{1}<{b}<{c}^{\mathrm{2}} <{a}^{\mathrm{10}} \\ $$ $${log}_{{a}} {b}+\mathrm{2}{log}_{{b}} {c}+\mathrm{5}{log}_{{c}} {a}=\mathrm{12} \\ $$ $${prove}\:{that} \\ $$ $$\mathrm{2}{log}_{{a}} {c}+\mathrm{5}{log}_{{c}} {b}+\mathrm{10}{log}_{{b}} {a}\geqslant\mathrm{21} \\ $$