Question

Solve 3 log6 2 + 2 log6 3 − log6 x = 2.

Answer 1

$D:x \ \textgreater \ 0\\\\3\log_62+2\log_63-\log_6x=2\\-----------------------\\Use:\\\log_ab=c\iff a^c=b\\\log_ab+\log_ac=\log_a(bc)\\\log_ab-\log_ac=\log_a(\frac{b}{c})\\\log_ab^c=c\log_ab\\-----------------------\\\log_62^3+\log_63^2-\log_6x=2\\\\\log_68+\log_69-\log_6x=\log_66^2\\\\\log_6(8\cdot9)-\log_6x=\log_636\\\\\log_672-\log_6x=\log_636\\\\\log_6\frac{72}{x}=\log_636\iff\dfrac{72}{x}=36\\\\\dfrac{x}{72}=\dfrac{1}{36}\ \ \ \ \ |multiply\ both\ sides\ by\ 72\\\\x=\dfrac{72}{36}\\\\\boxed{x=2}\in D$