Solve the inequality $\sin x > \ln x$

Question

How do I solve $\sin x > \ln x$?

I get that we need to solve it graphically. The part of the graph where the graph of $\sin x$ is above the graph of $\ln x$. But to state the proper solution set we first need to find out where these two graph(s) intersect . So we need to solve $\sin x = \ln x$. This is where I am stuck. If $C$ be the point where these two graphs intersect then I can write solution set as $x$ belongs to $(C, -\infty)$.

Answer 1

If you're permitted to use computers:

NMinimize[Abs[Sin[x] - Log[x]], x]

{1.05365*10^-11, {x -> 2.21911}}

If you're going to plot these functions (surely with a computer), then you're already using a computer, and should be able to use other computer functions as well.

Answer 2

You will have to use a numerical method such as the "Newton-Raphson method" (https://en.wikipedia.org/wiki/Newton%27s_method).

Answer 3

Since you look for the zero of function $$f(x)=\sin(x)-\log(x)$$ as said in comments and answers, you need a numerival method. Newton method would really be the simplest. Starting at a reasonable guess $x_0$, the method will update it according to $$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$$ which, in the present case, will be $$x_{n+1}=x_n+\frac{x_n (\log (x_n)-\sin (x_n))}{x_n \cos (x_n)-1}$$

Graphing, you will notice that the solution is "close" to $2$. So, let us use $x_0=2$ and use the formula to get the following iterates $$\left( \begin{array}{cc} n & x_n \\ 0 & 2.000000000 \\ 1 & 2.235934064 \\ 2 & 2.219185522 \\ 3 & 2.219107151 \\ 4 & 2.219107149 \end{array} \right)$$

We could continue for many iterations to "end" with $$x \approx 2.2191071489137460325957851882042901681753665565321$$ which is not recognized by any of the inverse symbolic calculators I tried.

Answer 4

Sure.

$$C := \lim_{n \rightarrow \infty} \left(x \mapsto \left[x + \frac{ x(\log x - \sin x)}{x \cos x - 1}\right]\right)^n(2)$$

Then $x \in (0, C)$. Yes, that's Newton's method, essentially, but it has a point: don't get wedded into thinking there's only one 'right' way to express the solution to an equation. Cobbling it together as a finite composite of some arbitrary set of "allowable" functions and constants is not the only way and even if you can do it, it is not necessarily the most illuminating way, either.

Heck, in this case, the equation itself might be the best representation, being the most transparent for the problem:

$C$ is the unique solution of $$\sin C = \ln C$$

Provided, you can prove the uniqueness, of course (which isn't too difficult). Yeah, that may "seem" underwhelming, but that's psychology, not maths.

Sadly, the common ways maths is taught (whether in school, through books and other reference materials, or both) create this impression, but the impression is wrong.