Consider the example $$\sin x=\log [x]$$ where $[\,·\,]$ represents Greatest Integer Function. It is a miscellaneous equation, and I have been told that the only way to solve it is to draw the graphs of both the functions and find the points of intersection. But that is very timeconsuming.
Is there any other way of solving miscellaneous equations or something which can predict the number of solutions at least, or is it that these equations cannot be solved algebraically?
In this specific case, you can solve it by figuring out what the value of $y$ could be if $\sin x=\log\lfloor x\rfloor=y$. Since $\sin x=y$ we must have $-1\leq y\leq 1$, and since $\log\lfloor x\rfloor=y$, $y$ must be the log of an integer. Assuming you mean natural log, this integer must be $1$ or $2$ (because its log is at most $1$).
Now try to solve the simultaneous equations for each possible value of $y$.