log2-X-log-2-2X-log-2-4X-X-

Question Number 130168 by Adel last updated on 23/Jan/21

\log \underset{X}{2} \cdot \log \underset{2 X}{2} = \log \underset{4 X}{2} X = ?

Commented by liberty last updated on 23/Jan/21

do you meant \log_{x} (2) . \log_{2 x} (2) = \log_{4 x} (2) ?

Answered by liberty last updated on 23/Jan/21

\frac{1}{\log_{2} (x)} . \frac{1}{\log_{2} (2 x)} = \frac{1}{\log_{2} (4 x)}

2 + \log_{2} (x) = \log_{2} (x) [1 + \log_{2} (x)]

let \log_{2} (x) = p

\Rightarrow 2 + p = p + p^{2}; p^{2} = 2 \to {\begin{cases} p = \sqrt{2} \\ p = - \sqrt{2} \end{cases}

\Rightarrow \log_{2} (x) = \pm \sqrt{2}

\Rightarrow x = 2^{\pm \sqrt{2}}; x = {\begin{cases} 2^{\sqrt{2}} \\ \frac{1}{2^{\sqrt{2}}} \end{cases}

Commented by Adel last updated on 23/Jan/21

thanks sir

Answered by MJS_new last updated on 23/Jan/21

\frac{\ln 2}{\ln x} \times \frac{\ln 2}{\ln 2 x} = \frac{\ln 2}{\ln 4 x}

t = \ln x

\frac{\ln 2}{t} \times \frac{\ln 2}{t + \ln 2} = \frac{\ln 2}{t + 2 \ln 2}

t (t + \ln 2) = \ln 2 (t + 2 \ln 2)

t^{2} = 2 {(\ln 2)}^{2}

t = \pm \sqrt{2} \ln 2

x = 2^{- \sqrt{2}} \lor x = 2^{\sqrt{2}}

Commented by liberty last updated on 23/Jan/21

yes \dots my answer correct . .

Commented by Adel last updated on 23/Jan/21

tanks sir

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