Kaila Branch

2022-09-24

no. and nature of roots of ${x}^{\frac{3}{4}({\mathrm{log}}_{2}x{)}^{2}+{\mathrm{log}}_{2}x-\frac{5}{4}}=\sqrt{2}$

The given equation is

${x}^{\frac{3}{4}({\mathrm{log}}_{2}x{)}^{2}+{\mathrm{log}}_{2}x-\frac{5}{4}}=\sqrt{2}$

I took ${\mathrm{log}}_{2}x$

and then rewrote the given equation as

${x}^{3{t}^{2}+4t-5}=\sqrt{2}$

But I don't know what to do after this. How will I find the nature and no. of roots?

The given equation is

${x}^{\frac{3}{4}({\mathrm{log}}_{2}x{)}^{2}+{\mathrm{log}}_{2}x-\frac{5}{4}}=\sqrt{2}$

I took ${\mathrm{log}}_{2}x$

and then rewrote the given equation as

${x}^{3{t}^{2}+4t-5}=\sqrt{2}$

But I don't know what to do after this. How will I find the nature and no. of roots?

Bridger Hall

Beginner2022-09-25Added 7 answers

${x}^{\frac{3}{4}({\mathrm{log}}_{2}x{)}^{2}+{\mathrm{log}}_{2}x-\frac{5}{4}}=\sqrt{2}$

${\mathrm{log}}_{2}{x}^{\frac{3}{4}({\mathrm{log}}_{2}x{)}^{2}+{\mathrm{log}}_{2}x-\frac{5}{4}}={\mathrm{log}}_{2}\sqrt{2}$

${\mathrm{log}}_{2}x(\frac{3}{4}({\mathrm{log}}_{2}x{)}^{2}+{\mathrm{log}}_{2}x-\frac{5}{4})=\frac{1}{2}$

$t={\mathrm{log}}_{2}x$

$3{t}^{3}+4{t}^{2}-5t-2=0$

${t}_{1}=1.$

Can you finish?

${\mathrm{log}}_{2}{x}^{\frac{3}{4}({\mathrm{log}}_{2}x{)}^{2}+{\mathrm{log}}_{2}x-\frac{5}{4}}={\mathrm{log}}_{2}\sqrt{2}$

${\mathrm{log}}_{2}x(\frac{3}{4}({\mathrm{log}}_{2}x{)}^{2}+{\mathrm{log}}_{2}x-\frac{5}{4})=\frac{1}{2}$

$t={\mathrm{log}}_{2}x$

$3{t}^{3}+4{t}^{2}-5t-2=0$

${t}_{1}=1.$

Can you finish?

Darius Miles

Beginner2022-09-26Added 3 answers

I'd try to take ${\mathrm{log}}_{2}$ of the whole expression and the solve with respect to $t={\mathrm{log}}_{2}x$

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