**
Logarithmic method:**

or.

So finding the logarithm of that number square of the number can be derived.

**
Newton Raphson method:**

Given
a function *ƒ*(*x*) and its derivative *ƒ* '(*x*),
we begin with a first guess *x*_{0}. Provided the function is
reasonably well-behaved a better approximation *x*_{1} is

Geometrically,
x_{1} is the intersection point of the tangent line
to the graph of f, with the x-axis. The process is repeated until a
sufficiently accurate value is reached:

The
idea of the method is as follows: one starts with an initial guess which is
reasonably close to the true root, then the function is approximated by its tangent line
(which can be computed using the tools of calculus),
and one computes the *x*-intercept of this tangent line (which is easily
done with elementary algebra). This *x*-intercept will typically be a
better approximation to the function's root than the original guess, and the
method can be iterated.

Suppose
*ƒ* : [*a*, *b*] → R is a differentiable
function defined on the interval [*a*, *b*] with
values in the real numbers R.
The formula for converging on the root can be easily derived. Suppose we have
some current approximation *x _{n}*. Then we can derive the formula
for a better approximation,

That is

Here,
*f* ' denotes the derivative of
the function *f*. Then by simple algebra we can derive

We
start the process off with some arbitrary initial value *x*_{0}.
(The closer to the zero, the better. But, in the absence of any intuition about
where the zero might lie, a "guess and check" method might narrow the
possibilities to a reasonably small interval by appealing to the intermediate value theorem.) The method will usually converge,
provided this initial guess is close enough to the unknown zero, and that *ƒ'*(*x*_{0})
≠ 0. Furthermore, for a zero of multiplicity 1, the convergence is at least
quadratic (see rate of convergence) in a neighborhood of the zero, which intuitively
means that the number of correct digits roughly at least doubles in every step.

** **