3^x = x^5 find x. i need to know how to do it, not just the answer. thanks..?

Answer 1

10.85138295870749...
1.343345558175514...
are the two solutions to this equation.

You could use an numerical approximation technique such as :
(1) interval bisection method
(2) secant method
(3) Newton-Raphson method
to find the values.

{ Newton will yield:
x(new) = x + (x^5 - 3^x)/[ ln(3)* 3^x - 5x^4 ]
as the generating equation.
start with x=1
calculate x(new), then plug this number back into the equation for x to find the next x(new)
repeat this and the sequence of x values will converge to the solution of the equation. The first 8 iterations are:
2.17359656484922
1.79597859258795
1.53548131479625
1.39216406941418
1.34739838877467
1.34337620387288
1.34334555994314
1.34334555817551

You will have to start with a value closer to 10.85... (say 10 or 11) to generate the other root.}


x can be also be expressed using the Lambert W-function, which is the inverse of the function f(x) = xe^x.

3^x = x^5
3^(1/5) = x^(1/x)
3^(-1/5) = (1/x)^(1/x)
let y = 1/x and c = 3^(-1/5)
y^y = c
y*ln(y) = ln(c)
let z = ln(y); or y = e^z
z*e^z = ln(c)
z = W(ln(c))
ln(y) = W(ln(c))
y = e^(W(ln(c)))
1/x = e^(W(ln(c)))
x = e^(-W(ln(c)) where c = 3^(-1/5) = 0.802741561...
or
x = e^(-W(-0.219722))

( note that W(n) is two valued for -1/e < n < 0 as in this case. )

W(-0.219722) = -2.3842925... or -0.12818555...
yielding the two solutions given.

Answer 2

A simple way to do it is to plot both functions and find where they intersect. A more complicated technique is to use some numerical method, such as Newton's rule, for finding the zero of 3^x-x^5.

Answer 3

Well I thought I could do this using a logarithm, but I hit a dead end. sorry

So I just graphed it on my TI-84.

I looked for the intersection and estimated it be x = 4/3 or x = 1.3433456.

I must admit that I have no idea how to symbolically manipulate the equation. Again sorry!

Answer 4

aprox = 1.343