Finding the distribution and density function of $Y = \log X$

Question

Let's $X$ be a random variable with exponential distribution.
That means that:

1. $F_X(t) = 1 - e^{1- \lambda t}$ which is the distribution function for $X$,
2. $f_X(t) = \lambda e^{- \lambda t}$ which is the density function for $X$.

Now let's define another random variable $Y = \log X$. Our task is to find it's distribution and density function.

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My attempt

Firstly I want to find the distribution function: $$F_{y}(t) := P(Y < t) = P(\log X < t) = P(X < e^t) = \int\limits_{0}^{e^t} \lambda e^{- \lambda x} \mbox{d}x$$ $$F_Y(t) = 1 - e^{- \lambda e^{t}}$$ Here's my question: how do I know what the integration limits should be? Is my reasoning correct?