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Question Number 70008 by Shamim last updated on 30/Sep/19

Find the value of x.  log_8 x + log_4 x + log_2 x = 11.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{x}. \\ $$$$\mathrm{log}_{\mathrm{8}} \mathrm{x}\:+\:\mathrm{log}_{\mathrm{4}} \mathrm{x}\:+\:\mathrm{log}_{\mathrm{2}} \mathrm{x}\:=\:\mathrm{11}. \\ $$

Answered by Maclaurin Stickker last updated on 30/Sep/19

log_2^3  x+log_2^2  x+log_2 x=11  (1/3)log_2 x+(1/2)log_2 x+log_2 x=11  ((1/3)+(1/2)+1)log_2 x=11  ((11)/6)log_2 x=11  log_2 x=6  2^6 =x  x=64

$${log}_{\mathrm{2}^{\mathrm{3}} } {x}+{log}_{\mathrm{2}^{\mathrm{2}} } {x}+{log}_{\mathrm{2}} {x}=\mathrm{11} \\ $$$$\frac{\mathrm{1}}{\mathrm{3}}{log}_{\mathrm{2}} {x}+\frac{\mathrm{1}}{\mathrm{2}}{log}_{\mathrm{2}} {x}+{log}_{\mathrm{2}} {x}=\mathrm{11} \\ $$$$\left(\frac{\mathrm{1}}{\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{2}}+\mathrm{1}\right){log}_{\mathrm{2}} {x}=\mathrm{11} \\ $$$$\frac{\mathrm{11}}{\mathrm{6}}{log}_{\mathrm{2}} {x}=\mathrm{11} \\ $$$${log}_{\mathrm{2}} {x}=\mathrm{6} \\ $$$$\mathrm{2}^{\mathrm{6}} ={x} \\ $$$${x}=\mathrm{64} \\ $$

Commented by Shamim last updated on 30/Sep/19

tnks a lot.

$$\mathrm{tnks}\:\mathrm{a}\:\mathrm{lot}. \\ $$

Commented by Shamim last updated on 30/Sep/19

plz explain with details-  log_2^3  x=(1/3)log_2 x  which law use for this??

$$\mathrm{plz}\:\mathrm{explain}\:\mathrm{with}\:\mathrm{details}- \\ $$$$\mathrm{log}_{\mathrm{2}^{\mathrm{3}} } \mathrm{x}=\frac{\mathrm{1}}{\mathrm{3}}\mathrm{log}_{\mathrm{2}} \mathrm{x} \\ $$$$\mathrm{which}\:\mathrm{law}\:\mathrm{use}\:\mathrm{for}\:\mathrm{this}?? \\ $$

Commented by Tony Lin last updated on 30/Sep/19

log_2^3  x=y  2^(3y) =x  log_2 x=3y  (1/3)log_2 x=y=log_2^3  x

$${log}_{\mathrm{2}^{\mathrm{3}} } {x}={y} \\ $$$$\mathrm{2}^{\mathrm{3}{y}} ={x} \\ $$$${log}_{\mathrm{2}} {x}=\mathrm{3}{y} \\ $$$$\frac{\mathrm{1}}{\mathrm{3}}{log}_{\mathrm{2}} {x}={y}={log}_{\mathrm{2}^{\mathrm{3}} } {x} \\ $$

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