What is the best approach to find the number of solutions to the equation?

Question

What is the best approach to find the number of solutions to the equation ?

$8[\log x]+6[e^x]=13+12[\cos x]$ ([.] denotes greatest integer function)

Is drawing their graphs and then checking for solutions the only approach?Is there any shorter approach?

Thanks.

Answer 1

Here $\lfloor x \rfloor $ is an Integer Quantity.

So $$\underbrace{8\lfloor \ln x \rfloor +6\lfloor e^x \rfloor}_{\bf{even\; integer}} = \underbrace{13+12\lfloor \cos x \rfloor}_{\bf{odd\; integer}} $$

So $\bf{L.H.S}$ is $\bf{even\; integer}$ and $\bf{R.H.S}$ is an $\bf{odd\; integer}$

So no real values of $x$

Answer 2

There could easily be infinite number of solutions (over the real numbers) to an equation involving integer parts. Maybe it is more interesting trying to count how many closed intervals are solutions to it. As an example, the equation:

$$[x] = [x^2]$$

would have solutions $\forall x \in [0,\sqrt 2]$, an unbearable number of real $x$:s, but only 1 interval.

Alternatively one could want to measure the total length of these intervals, or average truthfulness over an interval. I.e. if selecting $x$ according to some random distribution , then with what probability will equation be true?