How many roots does this exponential equation have? I have the following course assignment: Calculate the number of roots of the following equation: $ e^x = 6 \cdot \ln(x^2 + 1) $
What I have done so far:
$ f(x) = e^x - 6 \cdot \ln(x^2 + 1) $
I will look for the roots of the derivative and check where the function crosses OX as the function is continuous (Darboux):
$ f'(x) = e^x - \frac{12x}{x^2+1} $
$f'(x) \gt 0, x \leq 0 $
$f'(1) \lt 0$
$f'(2) \gt 0$
$ f(0) = 1, f(1) \lt 0, f(2) \lt 0 $
And there's the function graph:

I am missing the part (0,1) . Can somebody give me a hint? I guess it will look as I have drawn, but I cannot prove it.

You have most of the steps so far. I'll finish the problem off for you.
First, let's define our function $f(x)$ as:
$$f(x) = e^x - 6\ln(x^2+1)$$
Now, we can use the Intermediate Value Theorem to find our points that become the roots of this equation. We need to take the derivative in order to find them:
$$f'(x) = e^x - \frac{12x}{x^2 + 1}$$
Now, we set equal to $0$ to find our x values:
$$e^x - \frac{12x}{x^2 + 1} = 0$$
We could find the values algebraically, but consider this:
If $x < 0$, it follows that both our fraction and our exponential term will be positive. Therefore, $f'(x) > 0$ for all $x < 0$.
Next, we consider when $x > 0$. Now, initially the fraction term will "conquer" and become a greater value  than the exponential term for small values of $x$. However, as $x$ increases, the exponential term $e^x$ will ultimately increase at a higher rate over the fraction term.
Therefore, we have $2$ positive values of $x$ where $f'(x)  = 0$.

We can also say that $f'(x)$ changes from +, to -, and then +. This corresponds to $f(x)$ initially increasing, then decreasing, and then increasing again.
Now, in order to determine our roots. Let's see if we can find some trivial points.

For instance, if $x= 0$, our function f is $e^0 - 6\ln(1) = 1$
Again, if $x= -1$, our function f is $e^{-1} - 6\ln(2) < 0$, since $e^{-1}$ is a very small term compared to $6\ln 2$.
Since $f(x)$ is increasing on this interval, by the IVT we have one root in that range of $x$ values.
Next, let's try $x = 1$. $f(x)$ would then be $e^1 - 6\ln(2)$. $\ln 2$ is close to 2, so $6\ln 2$ compared to $e$ would be much greater and therefore $f(x) < 0$ in this case.
Therefore, by the IVT we have another root in that range of $x$ values.
Finally, we also mentioned that for larger values of $x$, the derivative $f'(x)$ would eventually increase faster than the rational term, so $f'(x) > 0$.
Therefore, $f(x)$ will eventually go through the x-axis again, so we have one more zero by IVT.

Therefore, we have $3$ zeros in total, looking at it analytically rather than algebraically. 
