How to solve an equation with a tangent divided by a logarithm?

Question

Here is an equation and I've never met this kind before. I would greatly appreciate your help. Maybe it's ridiculously simple and I overlook something?

$$-12=\frac{\tan(x+4)}{\log(x+0.25)}$$

Answer 1

To visualize why it has an infinite amount of solutions, add 12 to both sides and graph y=(the right side of the equation) on a graphing calculator.

Answer 2

For the above conclusion that can be done by hand consider: $$-12=\frac{\tan (x+4)}{\log (x +0.25)}$$ $$-12\log (x +0.25)=\tan (x+4)$$

Now using basic transformation rules of $f(x) \rightarrow Af(ax+b)+B$ learnt from high school (where $f=\log, \sin, \cos, \tan, polynomials$ and other elementary functions) you can sketch the LHS and RHS on the same paper

The number of intersections between $-12\log (x +0.25)$ and $\tan (x+4)$ is the number of solutions

Since $Range(LHS)=\mathbb{R},Domain(LHS)=(-0.25,\infty) \supset (0,\infty)$, has a root in the open domain and is monotonically increasing, it is guarantee to have at least one intersection with LHS. And since LHS is periodic, the same kind of behaviour will be repeated in the positive x direction every 4 units, result in infinitely many solutions