Find the total number of solutions of $ \log_e |\sin x| = - x^2 + 2x $ in 0 to $ \pi $ close interval.
So my book solves this problem by making graph. I am a high school student. I don't know how to make graph of such functions. I know the graph of log and I I know the graph of |sin| but I don't know how to solve this sin inside log.
If you can give a source which teaches that , that will solve my problem. I am not allowed to use calculator.
I tried solving the equations, I also tied finding range of both the sides to see if there is any intersection.
First, let's see the difference between the graphs of $|\sin(x)|$ and $\log|\sin(x)|$. Below, you can see what the log function does to the $|\sin(x)|$.
Now, let's talk about the computational side. I assume you may use a calculator. For graphing such functions, constructing a table would be a wise option. To obtain a value for each step, if you are unfamiliar, you can do the following. Assume you want to calculate the $y$ value for $x=\pi/2 \ (\approx 1.57)$ (you can also enter the following in one line in your calculator).
sin(pi/2) 1 log(ans) 0which is in line with the graph above. Now compute several of these points to graph the function yourself.
I assume you are now ready to start building your own table, and thereby also graphing your function? If not, or you want to check, I will provide a small table below.
Exercise for you: explain the symmetry within the table, and why did I choose my $x$ this way?