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Question Number 104835 by bemath last updated on 24/Jul/20
 { ((log _p (q) = x^2 )),((log _q (p^3 ) = x )) :}  find x
$$\begin{cases}{\mathrm{log}\:_{{p}} \left({q}\right)\:=\:{x}^{\mathrm{2}} }\\{\mathrm{log}\:_{{q}} \left({p}^{\mathrm{3}} \right)\:=\:{x}\:}\end{cases} \\ $$$${find}\:{x}\: \\ $$
Answered by john santu last updated on 24/Jul/20
 { ((q = p^x^2  →q^x  = (p^x^2  )^x )),((p^3  = q^x → p^3  = p^(x.x^2 ) )) :}  ⇒ x^3  = 3 ; x = (3)^(1/3)    (JS ⊛)
$$\begin{cases}{{q}\:=\:{p}^{{x}^{\mathrm{2}} } \rightarrow{q}^{{x}} \:=\:\left({p}^{{x}^{\mathrm{2}} } \right)^{{x}} }\\{{p}^{\mathrm{3}} \:=\:{q}^{{x}} \rightarrow\:{p}^{\mathrm{3}} \:=\:{p}^{{x}.{x}^{\mathrm{2}} } }\end{cases} \\ $$$$\Rightarrow\:{x}^{\mathrm{3}} \:=\:\mathrm{3}\:;\:{x}\:=\:\sqrt[{\mathrm{3}}]{\mathrm{3}}\:\:\:\left({JS}\:\circledast\right)\: \\ $$
Answered by Dwaipayan Shikari last updated on 24/Jul/20
log_p (q)=x^2   3log_q (p)=x  3log_q p.log_p q=x^3   3=x^3   x=(3)^(1/3)
$${log}_{{p}} \left({q}\right)={x}^{\mathrm{2}} \\ $$$$\mathrm{3}{log}_{{q}} \left({p}\right)={x} \\ $$$$\mathrm{3}{log}_{{q}} {p}.{log}_{{p}} {q}={x}^{\mathrm{3}} \\ $$$$\mathrm{3}={x}^{\mathrm{3}} \\ $$$${x}=\sqrt[{\mathrm{3}}]{\mathrm{3}} \\ $$

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