4 ms·
This is easily solved using the Lambert W function: Observe: W(x * e ^ x) = x x ^ x = y ln(x ^ x) = ln(y) ln(x) * x = ln(y) ln(x) * e ^ ln(x)
by youareawesome 7y ago
This is easily solved using the Lambert W function:
Observe: W(x * e ^ x) = x
x ^ x = y
ln(x ^ x) = ln(y)
ln(x) * x = ln(y)
ln(x) * e ^ ln(x) = ln(y)
W(ln(x) * e ^ ln(x)) = W(ln(y))
ln(x) = W(ln(y))
x = e^W(ln(y))
https://en.wikipedia.org/wiki/Lambert_W_function https://en.wikipedia.org/wiki/Lambert_W_function
- throwlaplace 7y agoThat's pretty asinine since the W function is that inverse (basically). You're just saying "this is easily solved by easily solving". Before anyone jumps on me: I'm perfectly familiar and comfortable with functions that aren't defined in closed form.
- paulddraper 7y agoTo be fair, this is much of math. The ratio of right triangle legs is "easily solved" by tan(x). But that's magical/cheating to someone who hasn't studied trigonometry. Lambert-W is a well known function with known approximation methods; once you reduce your problem to it (in this case, trivially), you can lean on that knowledge from others.
- knzhou 7y agoThat's true for almost any nontrivial transcendental algebraic or differential equation. Special functions to solve special cases are the norm. We could repeat the same discussion at a lower level. Suppose you tell a bright middle schooler about the basics of integration, and the power rule. Then they ask, "So what's the integral of 1/x? It can't be x^0/0..." "That's a special case. It's a function called "natural logarithm", ln(x)." "But what's the definition of that weird function?" "It's defined to be the integral of 1/x." [0] "That's pretty asinine..." [0] https://en.wikipedia.org/wiki/Natural_logarithm#Definitions https://en.wikipedia.org/wiki/Natural_logarithm#Definitions