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Question Number 215232 by Mingma last updated on 01/Jan/25

Answered by devdutt last updated on 01/Jan/25

ln 3(√5) + ln x log _(45) x = 2ln x ⇒ ln 3(√5) + ((ln^2 x)/(ln 45)) = 2ln x ⇒ ln^2 x − 2ln 45 ln x + ln 3(√5) ln 45 = 0 ⇒ ln x = ((2ln 45 ± (√(4ln^2 45 − 4ln 3(√5) ln 45)))/2) = ln 45 ± (√(ln^2 45 − ln 3(√5) ln 45)) ⇒ x = 45e^(±(√(ln^2 45 − ln 3(√5) ln 45))) product of roots = 45^2 = 2025

$$\mathrm{ln}\:\mathrm{3}\sqrt{\mathrm{5}}\:+\:\mathrm{ln}\:{x}\:\mathrm{log}\:_{\mathrm{45}} {x}\:=\:\mathrm{2ln}\:{x}\: \\ $$$$\Rightarrow\:\mathrm{ln}\:\mathrm{3}\sqrt{\mathrm{5}}\:+\:\frac{\mathrm{ln}\:^{\mathrm{2}} {x}}{\mathrm{ln}\:\mathrm{45}}\:=\:\mathrm{2ln}\:{x}\: \\ $$$$\Rightarrow\:\mathrm{ln}\:^{\mathrm{2}} {x}\:−\:\mathrm{2ln}\:\mathrm{45}\:\mathrm{ln}\:{x}\:+\:\mathrm{ln}\:\mathrm{3}\sqrt{\mathrm{5}}\:\mathrm{ln}\:\mathrm{45}\:=\:\mathrm{0} \\ $$$$\Rightarrow\:\mathrm{ln}\:{x}\:=\:\frac{\mathrm{2ln}\:\mathrm{45}\:\pm\:\sqrt{\mathrm{4ln}\:^{\mathrm{2}} \mathrm{45}\:−\:\mathrm{4ln}\:\mathrm{3}\sqrt{\mathrm{5}}\:\mathrm{ln}\:\mathrm{45}}}{\mathrm{2}}\:=\:\mathrm{ln}\:\mathrm{45}\:\pm\:\sqrt{\mathrm{ln}\:^{\mathrm{2}} \mathrm{45}\:−\:\mathrm{ln}\:\mathrm{3}\sqrt{\mathrm{5}}\:\mathrm{ln}\:\mathrm{45}} \\ $$$$\Rightarrow\:{x}\:=\:\mathrm{45}{e}^{\pm\sqrt{\mathrm{ln}\:^{\mathrm{2}} \mathrm{45}\:−\:\mathrm{ln}\:\mathrm{3}\sqrt{\mathrm{5}}\:\mathrm{ln}\:\mathrm{45}}} \\ $$$$ \\ $$$${product}\:{of}\:{roots}\:=\:\mathrm{45}^{\mathrm{2}} \:=\:\mathrm{2025} \\ $$$$ \\ $$

Commented by Mingma last updated on 02/Jan/25

Perfect ��

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