real analysis – Searching the roots of a self-consistent transcendental equation

I have the equation

$$M = c_1 + c_2M – c_3Tlnleft(left|frac{e^{(c_4M + c_5)/T}-1}{e^{(c_6M + c_5)/T}-1}right|right)$$

where $c_1, dots, c_6$ are constants.

According to the following graph, the equation should have roots.

enter image description here

My question is, how to find analytically the cut off point $T_c$ after which the equation has no roots?