What is the value of x at x=2
Answers
Step-by-step explanation:
There are three values of x : - 0.76666029 .. 2 , and 4.
===
Given 2ˣ = x² To solve for x.
Take logarithms on both sides.
x Log 2 = 2 Log x as Log xⁿ = n Log x
Well to solve the exponential equations is not easy. We can use the graphical means to solve some of this type of questions.
See diagram for the graphs of 2^x and x^2.
x^2 and 2^x meet at x = 2 and x = 4.
For x > 4, 2^x is always more than x^2. So there are only two solutions.
x = 2 and x = 4.
Perhaps we can use numerical approximation methods like Newton Raphson or Range Kutta methods. These are iteration methods.
Another Iteration method:
f(x) = 2^x - x^2
f'(x) = 2^x * Ln 2 - 2 x
f(x) = f(1) + f '(1) * (x - 1)
= 1 + (Ln4 - 2) (x - 1) = - 0.6137 x + 1.6137
We want f(x) = 0.
=> x1 = 1.6137/0.6137 = 2.6
Now substitute x1 = 2.6 and find
f(x) = 0 = f(2.6) + f '(2.6) * (x2 - 2.6)
Find next value of x as x2.
Again write the equation 0 = f(x2) + f '(x2) * (x3 - x2)
When the difference between successive values of x is small enough, we have the answer.
Showing the graph on the negative side also, or using the approximation methods on the negative x axis, we get x = -0.7666602 ..Also at which they are equal.