2logₓ2+0.5log(subscript y)3 = 5 3logₓ2-2log(subscript y)3 = 13 Solve for all real (x,y). ?

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Answer 1 · 2018-04-01T05:10:11

#x=root(3)2# and #y=1/sqrt3#

#2log_x2+0.5log_y3=5#

i.e. #(2log2)/logx+log3/(2logy)=5# .........................(A)

#3log_x2-2log_y3=13#

i.e. #(3log2)/logx-(2log3)/logy=13# .........................(B)

Multiplying (A) by #3# and (B) by #2# and subtracting latter from former, we get

#(6log2)/logx-(6log2)/logx+(3log3)/(2logy)+(4log3)/logy=15-26#

or #(11log3)/(2logy)=-11#

i.e. #logy=-log3/2=log3^(-1/2)#

and #y=3^(-1/2)=1/sqrt3#

Putting #2logy=-log3# in (A), we get

#(2log2)/logx-log3/log3=5#

or #(2log2)/logx=6#

i.e. #logx=log2/3=log2^(1/3)# i.e. #x=root(3)2#