Solve-for-x-x-2-log-2-x-8-

Question Number 4399 by alib last updated on 22/Jan/16

S o l v e f o r x

x^{2 l o g_{2} x} = 8

Commented by Rasheed Soomro last updated on 20/Jan/16

x^{2 l o g_{2} x} = 8

x^{{l o g}_{2} x^{2}} = 8

L e t 8 = x^{y}

{l o g}_{2} 8 = {y l o g}_{2} x

3 = y {l o g}_{2} x

y = 3 / {l o g}_{2} x

S o 8 = x^{3 / {l o g}_{2} x}

x^{{l o g}_{2} x^{2}} = x^{3 / {l o g}_{2} x}

{l o g}_{2} x^{2} = 3 / {l o g}_{2} x

({l o g}_{2} x^{2}) ({l o g}_{2} x) = 3

2 {l o g}_{2} x {l o g}_{2} x = 3

{({l o g}_{2} x)}^{2} = \frac{3}{2}

{l o g}_{2} x = \pm \sqrt{\frac{3}{2}}

x = 2^{\pm \sqrt{1.5}}

x \approx 2.3371, 0.42787

Commented by Yozzii last updated on 20/Jan/16

x^{{l o g}_{2} x^{2}} = 8

{{l o g}_{2} x^{2}} (l n x) = l n 8

\frac{{l n x}^{2}}{l n 2} = \frac{l n 8}{l n x} (c h a n g e o f b a s e)

\frac{2 l n x}{l n 2} = \frac{l n 8}{l n x}

2 {(l n x)}^{2} = l n 2 \times l n 8

l n x = \pm \sqrt{\frac{l n 2 \times l n 8}{2}}

x = e x p (\pm \sqrt{0.5 l n 2 \times l n 8}) \approx 2.3371, 0.4279

Answered by Rasheed Soomro last updated on 22/Jan/16

x^{2 l o g_{2} x} = 8

{l o g}_{2} (x^{2 {l o g}_{2} x}) = {l o g}_{2} 8

2 {l o g}_{2} x {l o g}_{2} x = 3

{({l o g}_{2} x)}^{2} = 3 / 2

{l o g}_{2} x = \pm \sqrt{3 / 2}

x = 2^{\pm \sqrt{1.5}}

x = 2.33714, 0.427873