How do you solve ${e^x} = 0$ ?

Hint:
In this sum the student has to take log on both the sides and then use the properties of Log to solve the numerical. After taking the logarithm the student has to bring all the constants on one side and the variable on the other. Last step is to input the value of log to get the final answer. Though the final answer for this particular sum is not defined as a logarithm of $0$ is not defined. There is no number that satisfies the equation when x equals to any value.

Complete step by step solution:
Considering all the logarithms have the base $0$.
In this particular sum we will be using the property $\log {a^b} = b\log a$
Let us take logs on both the sides in the given sum. After adding log the next step is
$\log ({e^x}) = \log 0..........(1)$
Utilizing the property of Logs - $\log {a^b} = b\log a$, we have
$x\log (e) = \log 0..........(2)$
We have made no assumptions regarding the base of the logarithm, therefore using base $e$ logarithms( in conjunction with the calculator)
$x{\log _e}e = \log 0..........(4)$
As we know that there is no such value which satisfies this equation. We can say that the answer to this particular sum is not defined.

Therefore value of $x$ is not defined for the equation ${e^x} = 0$

Note:
Whenever the student is confused whether to use indices or Logarithm, he /she should always go for logarithm. This is because the final answer would be the same, only the difference in the number of steps would be more in the case of logarithm approach. Students should always follow the logarithm method as it makes the sum easy to solve and without creating any complications, unlike indices where the student might forget the property of indices. He/she should always keep in mind while solving logarithmic sums that $\log (1) = 0$& $\log (0)$ is not defined.