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Question Number 129451 by bemath last updated on 15/Jan/21

log _(12) ((√x) + (x)^(1/4) ) = log _9 ((√x) ) x = ?

$$\:\mathrm{log}\:_{\mathrm{12}} \left(\sqrt{\mathrm{x}}\:+\:\sqrt[{\mathrm{4}}]{\mathrm{x}}\:\right)\:=\:\mathrm{log}\:_{\mathrm{9}} \left(\sqrt{\mathrm{x}}\:\right)\: \\ $$$$\:\mathrm{x}\:=\:? \\ $$

Answered by liberty last updated on 16/Jan/21

log _(12) ((√x) + (x)^(1/4) ) = log _3 ((x)^(1/4) ) let x = t^4 ⇒((ln (t^2 +t))/(ln 12)) = ((ln (t))/(ln 3)) ln 3. ln (t^2 +t) = ln 12.ln (t) ln 3. ln (t^2 +t)=(ln 3+ln 4).ln (t) holds for t = 3 ; then x = 3^4 = 81

$$\:\mathrm{log}\:_{\mathrm{12}} \left(\sqrt{\mathrm{x}}\:+\:\sqrt[{\mathrm{4}}]{\mathrm{x}}\:\right)\:=\:\mathrm{log}\:_{\mathrm{3}} \left(\sqrt[{\mathrm{4}}]{\mathrm{x}}\:\right) \\ $$$$\:\mathrm{let}\:\mathrm{x}\:=\:\mathrm{t}^{\mathrm{4}} \:\Rightarrow\frac{\mathrm{ln}\:\left(\mathrm{t}^{\mathrm{2}} +\mathrm{t}\right)}{\mathrm{ln}\:\mathrm{12}}\:=\:\frac{\mathrm{ln}\:\left(\mathrm{t}\right)}{\mathrm{ln}\:\mathrm{3}} \\ $$$$\:\mathrm{ln}\:\mathrm{3}.\:\mathrm{ln}\:\left(\mathrm{t}^{\mathrm{2}} +\mathrm{t}\right)\:=\:\mathrm{ln}\:\mathrm{12}.\mathrm{ln}\:\left(\mathrm{t}\right) \\ $$$$\:\mathrm{ln}\:\mathrm{3}.\:\mathrm{ln}\:\left(\mathrm{t}^{\mathrm{2}} +\mathrm{t}\right)=\left(\mathrm{ln}\:\mathrm{3}+\mathrm{ln}\:\mathrm{4}\right).\mathrm{ln}\:\left(\mathrm{t}\right) \\ $$$$\:\mathrm{holds}\:\mathrm{for}\:\mathrm{t}\:=\:\mathrm{3}\:;\:\mathrm{then}\:\mathrm{x}\:=\:\mathrm{3}^{\mathrm{4}} \:=\:\mathrm{81}\: \\ $$$$ \\ $$

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