If 1,log_{10}(4^{x}-2) and log_{10}(4^{x}+fra | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Given that $1$,$\log_{10}(4^{x}-2)$,and $\log_{10}(4^{x}+\frac{18}{5})$ are in arithmetic progression,we have:$2 \log_{10}(4^{x}-2) = 1 + \log_{10

Vedclass · Accepted Answer

$2$

Vedclass · Answer

(D) Given that $1$,$\log_{10}(4^{x}-2)$,and $\log_{10}(4^{x}+\frac{18}{5})$ are in arithmetic progression,we have:$2 \log_{10}(4^{x}-2) = 1 + \log_{10}(4^{x}+\frac{18}{5})$$\log_{10}(4^{x}-2)^{2} = \log_{10}(10) + \log_{10}(4^{x}+\frac{18}{5})$$(4^{x}-2)^{2} = 10(4^{x}+\frac{18}{5})$$(4^{x})^{2} - 4(4^{x}) + 4 = 10(4^{x}) + 36$$(4^{x})^{2} - 14(4^{x}) - 32 = 0$Let $y = 4^{x}$,then $y^{2} - 14y - 32 = 0$$(y-16)(y+2) = 0$Since $4^{x} > 0$,we have $4^{x} = 16$,which implies $x = 2$.Now,substitute $x=2$ into the determinant:$\left|\begin{array}{ccc} 2(2-\frac{1}{2}) & 2-1 & 2^{2} \ 1 & 0 & 2 \ 2 & 1 & 0 \end{array}ight| = \left|\begin{array}{ccc} 3 & 1 & 4 \ 1 & 0 & 2 \ 2 & 1 & 0 \end{array}ight|$$= 3(0-2) - 1(0-4) + 4(1-0)$$= -6 + 4 + 4 = 2$.

If $1$,$\log_{10}(4^{x}-2)$ and $\log_{10}(4^{x}+\frac{18}{5})$ are in arithmetic progression for a real number $x$,then the value of the determinant $\left|\begin{array}{ccc} 2(x-\frac{1}{2}) & x-1 & x^{2} \\ 1 & 0 & x \\ x & 1 & 0 \end{array}\right|$ is equal to ...... .

Similar Questions

The equation of the line joining the origin $(0, 0)$ to the point $(-4, 5)$ is:

The value of the determinant $\left| \begin{array}{ccc} 1 & \cos(\alpha - \beta) & \cos \alpha \\ \cos(\alpha - \beta) & 1 & \cos \beta \\ \cos \alpha & \cos \beta & 1 \end{array} \right|$ is

If $A = \begin{bmatrix} 1 & 0 & 1 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{bmatrix}$,then $\det A$ =

If $-9$ is a root of the equation $\left| \begin{array}{ccc} x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x \end{array} \right| = 0$,then the other two roots are:

If $5$ is one root of the equation $\left| \begin{array}{ccc} x & 3 & 7 \\ 2 & x & -2 \\ 7 & 8 & x \end{array} \right| = 0$,then the other two roots of the equation are:

Vedclass Test Series

Exam Paper Generator

Online Exam Module