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Question Number 12489 by Joel577 last updated on 23/Apr/17

log_2 x + log_(1/x) (1/2) ≥ 0 x = ?

$$\mathrm{log}_{\mathrm{2}} \:{x}\:+\:\mathrm{log}_{\mathrm{1}/{x}} \:\frac{\mathrm{1}}{\mathrm{2}}\:\geqslant\:\mathrm{0} \\ $$$${x}\:=\:? \\ $$

Answered by mrW1 last updated on 23/Apr/17

((ln x)/(ln 2))+((ln ((1/2)))/(ln ((1/x))))≥0 ((ln x)/(ln 2))+((ln 2)/(ln x))≥0 (((ln x)^2 +(ln 2)^2 )/((ln 2)(ln x)))≥0 (ln 2)(ln x)>0 ln x>0 ⇒x>1

$$\frac{\mathrm{ln}\:{x}}{\mathrm{ln}\:\mathrm{2}}+\frac{\mathrm{ln}\:\left(\frac{\mathrm{1}}{\mathrm{2}}\right)}{\mathrm{ln}\:\left(\frac{\mathrm{1}}{{x}}\right)}\geqslant\mathrm{0} \\ $$$$\frac{\mathrm{ln}\:{x}}{\mathrm{ln}\:\mathrm{2}}+\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{ln}\:{x}}\geqslant\mathrm{0} \\ $$$$\frac{\left(\mathrm{ln}\:{x}\right)^{\mathrm{2}} +\left(\mathrm{ln}\:\mathrm{2}\right)^{\mathrm{2}} }{\left(\mathrm{ln}\:\mathrm{2}\right)\left(\mathrm{ln}\:{x}\right)}\geqslant\mathrm{0} \\ $$$$\left(\mathrm{ln}\:\mathrm{2}\right)\left(\mathrm{ln}\:{x}\right)>\mathrm{0} \\ $$$$\mathrm{ln}\:{x}>\mathrm{0} \\ $$$$\Rightarrow{x}>\mathrm{1} \\ $$

Commented by Joel577 last updated on 24/Apr/17

thank you very much

$${thank}\:{you}\:{very}\:{much} \\ $$

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