Exponential equation with a negative exponent

Question

From the first sight, this equation:

$\exp(-2at)=-\exp(-2bt)$

has no solution.

However, Worfram Mathematica clams, it exists. I am wondering, what is the most common to solve it: perhaps, Taylor expansion? Minus in from of the second exponent forbids using the log-mathod. Thank you very much in advance.

Answer 1

Since $\exp$ is a positive function, $-\exp$ will be negative, hence $\exp(\text{whatever}_1)=-\exp (\text{whatever}_2)$ has no real solutions.

Answer 2

Hint: $$e^{2bt-2at}=-1=e^{i\pi}$$

Answer 3

$\displaystyle e^{2at}=-e^{2bt}\Rightarrow {e^{2(a-b)t}}=-1\Rightarrow {e^{2(a-b)t}}=e^{i\pi}$