Question

Number of solution(s) possible for equation $\sin \left(x\right)={\log }_{10}\left(x\right)$ is/are

• A. 1
• B. 2
• C. 3
• D. more than 3

Answer 1

The correct option is C 3

Solution to equation $f\left(x\right)=0$ is the value of x which satisfies that equation.
For the given equation, we have to find the number of values of x at which $\sin \left(x\right)={\log }_{10}\left(x\right)$
This means if we plot the graph of functions $\sin \left(x\right)\mathrm{and}{\log }_{10}\left(x\right),$
the points at which both intersect each other are the solutions to the equation $\sin \left(x\right)={\log }_{10}\left(x\right).$
Below shown is the plot of both curves on X-Y plane.

We know${\log }_a\left(a\right)=1\Rightarrow {\log }_{10}\left(10\right)=1i.e.,\mathrm{At}x=10,{\log }_{10}x=1\mathrm{For}x>10,y={\log }_{10}x\mathrm{will}\mathrm{always}\mathrm{be}\mathrm{greater}\mathrm{than}1.\mathrm{Also},\mathrm{sinx}\mathrm{lies}\mathrm{between}-1\mathrm{and}+1.\mathrm{So},\sin \left(x\right)\mathrm{and}{\log }_{10}x\mathrm{will}\mathrm{never}\mathrm{intersect}\mathrm{when}x>10.\mathrm{Hence},\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{solution}\mathrm{possible}\mathrm{for}\mathrm{the}\mathrm{given}\mathrm{equation}\mathrm{is}\mathrm{three}\mathrm{as}\mathrm{it}\mathrm{intersects}\mathrm{only}\mathrm{thrice}.$