Question Number 196249 by mathlove last updated on 20/Aug/23
$${log}_{{a}} {x}=\mathrm{30} \\ $$$${log}_{{b}} {x}=\mathrm{70} \\ $$$${log}_{{ab}} {x}=? \\ $$
Answered by AST last updated on 20/Aug/23
$$\frac{{logx}}{{loga}}=\mathrm{30};\frac{{logx}}{{logb}}=\mathrm{70}\Rightarrow\frac{{log}\left({b}\right)+{log}\left({a}\right)}{{log}\left({x}\right)}=\frac{\mathrm{1}}{\mathrm{30}}+\frac{\mathrm{1}}{\mathrm{70}}=\frac{\mathrm{1}}{\mathrm{21}} \\ $$$${log}_{{ab}} {x}=\frac{{logx}}{{log}\left({ab}\right)}=\frac{\mathrm{1}}{\frac{{log}\left({b}\right)+{log}\left({a}\right)}{{log}\left({x}\right)}}=\mathrm{21} \\ $$