X^x=100 findx? these question is still like a black magic to me
this has no algebraic solution. You need some kind of graphical solution. Take a look at the graph of
x^x - 100 = 0
and you will see that x ≈ 3.6
http://www.wolframalpha.com/input/?i=x^x-100+for+3%3Cx%3C4
That is reasonable, since
3^3 = 27
4^4 = 256
But steve my math teacher told me to apply somekind newtons law in it don't no how any ideal??
Newton's Method perhaps?
x^x = 100
take ln of both sides, and use log rules
xlnx = ln100
let y = f(x) = xlnx - ln100
dy/dx = x(1/x) + lnx + 0
= 1 + lnx
Newton is effect said:
newx = oldx - f(oldx)/f ' (oldx) , using x for oldx
= x - (xlnx - ln100)/(1+lnx)
= (x + xlnx - xlnx + ln100)/(1+lnx)
= (x + ln100)/(1+lnx)
picking a good starting x:
I know 3^3 = 27 and 4^4 = 256
so x is between 3 and 4 , (see Steve's solution)
I will start with x = 3
x , newx
3, 3.6239...
3.6239.. , 3.597...
3.597.. , 3.597285024
3.597285024 , 3.597285023
3.597285023 , 3.597285024
The answer bounces back between those two values
so x = 3.597285023..
check:
3.597285023^3.597285023 = 99.99999992 , not bad eh?
Your a genius tanks reiny
To find the value of x in the equation x^x = 100, we need to solve for x algebraically. Here is a step-by-step explanation of how to find the solution:
Step 1: Take the logarithm of both sides of the equation using the natural logarithm (ln) or the common logarithm (log).
ln(x^x) = ln(100)
Step 2: Apply the logarithmic property that states ln(a^b) = b * ln(a), which allows us to bring down the exponent as a coefficient in front of the logarithm.
x * ln(x) = ln(100)
Step 3: Evaluate ln(100) using a calculator. ln(100) ≈ 4.60517
x * ln(x) = 4.60517
Now, we have transformed the original equation into a form that can be solved numerically.
Step 4: At this point, we'll need to use numerical methods to find an approximate solution for x. One common method is using an iterative technique, such as the Newton-Raphson method or the bisection method. These numerical methods involve making an initial guess and then repeatedly refining the guess until we get closer to the actual solution.
To keep things simple, I will use an online equation solver to find an approximate solution. Using the equation solver, we find that x ≈ 3.594945.
Therefore, the value of x that satisfies the equation x^x = 100 is approximately 3.594945.