The equation x3-4log2 x2-log2 x-5-4-2 has

Given, x^3/4(log_2 x)^2+log_2 x 5/4=√(2) ⇒3/4(log_2 x)^2+log_2x 5/4=log_x √(2) ⇒3/4(log_2 x)^2+log_2 x 5/4=1/2 log_2 x ⇒ 3(log_2 x)^3+4 (log_2 x)^2 5 (log_2 x ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Higher Order Equations

The equation ...

Question

The equation $x^{\frac{3}{4} (l o g_{2} x)^{2} + l o g_{2} x - \frac{5}{4}} = \sqrt{2}$ has

atleast one real solution

No worries! We‘ve got your back. Try BYJU‘S free classes today!

exactly three real solutions

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

exactly one irrational solution

No worries! We‘ve got your back. Try BYJU‘S free classes today!

complex roots

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is B exactly three real solutions
Given, $x^{\frac{3}{4} (l o g_{2} x)^{2} + l o g_{2} x - \frac{5}{4}} = \sqrt{2}$
$\Rightarrow \frac{3}{4} (l o g_{2} x)^{2} + l o g_{2} x - \frac{5}{4} = l o g_{x} \sqrt{2} \Rightarrow \frac{3}{4} (l o g_{2} x)^{2} + l o g_{2} x - \frac{5}{4} = \frac{1}{2 l o g_{2} x} \Rightarrow 3 (l o g_{2} x)^{3} + 4 (l o g_{2} x)^{2} - 5 (l o g_{2} x) - 2 = 0$
Put $l o g_{2} x = y$
$∴ 3 y^{3} + 4 y^{2} - 5 y - 2 = 0 \Rightarrow (y - 1) (y + 2) (3 y + 1) = 0 \Rightarrow y = 1, - 2, - \frac{1}{3} \Rightarrow l o g_{2} x = 1, - 2, - \frac{1}{3}$
$x = 2, \frac{1}{2^{\frac{1}{3}}}, \frac{1}{4}$

Suggest Corrections

Similar questions

Q. The equation

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

has-

Q. The equation

x^{\frac{3}{4} (l o g_{2} x)^{2} + l o g_{2} x - \frac{5}{4}} = \sqrt{2}

has

Solve the following equation for x
$x^{[\frac{3}{4} ({log}_{2} x)^{2} + {log}_{2} x - \frac{5}{4}]} = \sqrt{2}$

Q. The equation,

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

has

Q. The equation

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

has

Join BYJU'S Learning Program

The equation x34(log2x)2+log2x−54=√2 has

The correct option is B exactly three real solutionsGiven, x34(log2x)2+log2x−54=√2 ⇒34(log2x)2+log2x−54=logx√2⇒34(log2x)2+log2x−54=12log2x⇒3(log2x)3+4(log2x)2−5(log2x)−2=0 Put log2x=y ∴3y3+4y2−5y−2=0⇒(y−1)(y+2)(3y+1)=0⇒y=1,−2,−13⇒log2x=1,−2,−13 x=2,1213,14

The equation $x^{\frac{3}{4} (l o g_{2} x)^{2} + l o g_{2} x - \frac{5}{4}} = \sqrt{2}$ has