Today on my first day of break I found myself bored already and messing around with exponential equations.
A lot of very simple equations, such as a^b = b^a, are not solvable by typical algebra, although sometimes variants are. For example, it's possible if b is a known multiple of a... ie, x^(kx) = (kx)^x, solving for x in terms of k, you get x = k^(1/(k-1)). But most equations with exponentials, including things as simple as e^x = x (which has no real solution and only one complex solution, x = 0.31813150520476413531 - 1.3372357014306894*i) are just impossible with standard algebra unless you're clever or have a computer.
The point here is to not google/wolframalpha/maple/matlab these but instead try to figure them out yourselves, without using the kinds of non-elementary functions people have devised (ie, LambertW). For an example of what I mean, see the hint I posted for #1.
Roughly in order of difficulty...
[1]
e^y = x*y
Find y in terms of x, ie, find the function y(x)
All you need is one function
[2]
For a function f(x) = x^(x^(x^(x^(x^(...
which goes on to infinity,
find the derivative f ' (x)
And also find the values of x such that f(x) is finite.
[3]
Now introducing the "W" function, which solves this equation: z = W*e^W. So W(0) is the solution for W of 0 = W*e^W, i.e. the value is W(0) = 0. Other values are W(-pi/2) = i*pi/2, and W(-1/e) = -1, and W(1) is the omega constant (.56714).
Armed with this "non elementary" function, solve this analog of the quadratic equation for x:
Question
unreality
Today on my first day of break I found myself bored already and messing around with exponential equations.
A lot of very simple equations, such as a^b = b^a, are not solvable by typical algebra, although sometimes variants are. For example, it's possible if b is a known multiple of a... ie, x^(kx) = (kx)^x, solving for x in terms of k, you get x = k^(1/(k-1)). But most equations with exponentials, including things as simple as e^x = x (which has no real solution and only one complex solution, x = 0.31813150520476413531 - 1.3372357014306894*i) are just impossible with standard algebra unless you're clever or have a computer.
The point here is to not google/wolframalpha/maple/matlab these but instead try to figure them out yourselves, without using the kinds of non-elementary functions people have devised (ie, LambertW). For an example of what I mean, see the hint I posted for #1.
Roughly in order of difficulty...
[1]
e^y = x*y
Find y in terms of x, ie, find the function y(x)
[2]
For a function f(x) = x^(x^(x^(x^(x^(...
which goes on to infinity,
find the derivative f ' (x)
And also find the values of x such that f(x) is finite.
[3]
Now introducing the "W" function, which solves this equation: z = W*e^W. So W(0) is the solution for W of 0 = W*e^W, i.e. the value is W(0) = 0. Other values are W(-pi/2) = i*pi/2, and W(-1/e) = -1, and W(1) is the omega constant (.56714).
Armed with this "non elementary" function, solve this analog of the quadratic equation for x:
a*ln(ln(x)) + b*ln(x) + c*x = 0
(hint: try it with blnx + cx = 0 first)
[4]
Use W to invert these functions:
f(x) = x^x
f(x) = x^(x^(x^(x^(x^(x^(x^(x^(x^(x^(x^(x^(x^(x^(x^(x^(x^(x^(...
f(x) = x*ln(x)
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