Question

The number of solutions of the equation $x^{\left(\frac{\log x+7}{4}\right)}={10}^{\log x+1}$ is

• A. zero
• B. $1$
• C. $2$
• D. none of these

Answer 1

The correct option is C $2$

$x^{\frac{(logx+7)}{4}}={10}^{logx+1}$

taking log both sides we get

$\left(\frac{logx+7}{4}\right)logx=(logx+1)letlogx=t,x>o$(as domain of log is greater than 0)

$\left(\frac{t+7}{4}\right)t=t+1t^2+7t=4t+4t^2+3t-4=0(t+4)(t-1)=0t=1or-4t=1logx=1x={10}^{1}x=10,t=-4logx=-4x={10}^{-4}x=\frac{1}{{10}^4}$ so 2 solutions of x