ProductLog

To solve this problem, we need the use of the Lambert's W Function or the Product Log: $W(n)$

For a good explanation of its use, watch this video by BlackPenRedPen

First, we need to solve the two equations for $x$ and $y$ :

1) $xe^x = e^3$ is easy because we can just use the Lambert's W Function straight away:

$W(xe^x) = W(e^3)$ .

Because of the properties of the Lambert's W Function, we get:

$x = W(e^3)$

2) However, we need to do more work for solving:

$y(ln(y) - 1) = e^4$

(beforehand i thought that the '- 1' was inside the log but i realised that it wasn't)

We can write $y$ as $e^ln(y)$ , which we can use to give us:

$(ln(y) - 1)e^{ln(y)} = e^4$ .

We can adjust this to fit the format of the Lambert's W Function by multiplying by $e^{-1}$ on both sides. Once we neaten it up a bit, we have:

$(ln(y) - 1)e^{ln(y)-1} = e^3$

Finally! We can use the Lambert's W Function to get:

$ln(y)-1 = W(e^3)$

And after rearranging for y we get:

$y = e^{W(e^3) + 1}$

3) Perfect! Now we just need to find out what $ln(xy)$ is. We can plug $x$ and $y$ in to make:

$ln(W(e^3)e^{W(e^3) + 1})$

Looks like we can just use some log properties to separate the two. Here is what we have:

$ln(W(e^3)) + ln(e^{W(e^3) + 1})$

Before we get to the part of this that is pretty impressive, we can cancel the $ln$ and the $e$ out:

$ln(W(e^3)) + W(e^3) + 1$

Now here is the main event in my opinion. A property of the Lambert's W Function which was unknown to me until figuring out this problem.

It turns out that the product log and the natural log work together and form their own little property which I think is pretty cool.

If you have $ln(W(e^n))$ , it turns out that this is equal to $n - W(e^n)$ ! This took me by surprise!

Unfortunately I have not been able to research this at the time of writing but its something really cool.

Anyway, we can now use this property to get:

$3 - W(e^3) + W(e^3) + 1$

We can now cancel out the product logs to give us the wonderful answer of $4$ !

Note to the author of this problem: This was really quite nice and the twist at the end with this property made it even better!