Graphing y = sqrt(log2(x))

You have entered [src]

           ________
          / log(x) 
f(x) =   /  ------ 
       \/   log(2)

$$f{\left(x \right)} = \sqrt{\frac{\log{\left(x \right)}}{\log{\left(2 \right)}}}$$

f = sqrt(log(x)/log(2))

The graph of the function

The points of intersection with the X-axis coordinate

Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
$$\sqrt{\frac{\log{\left(x \right)}}{\log{\left(2 \right)}}} = 0$$
Solve this equation
The points of intersection with the axis X:

Analytical solution
$$x_{1} = 1$$
Numerical solution
$$x_{1} = 1$$

The points of intersection with the Y axis coordinate

The graph crosses Y axis when x equals 0:
substitute x = 0 to sqrt(log(x)/log(2)).
$$\sqrt{\frac{\log{\left(0 \right)}}{\log{\left(2 \right)}}}$$
The result:
$$f{\left(0 \right)} = \tilde{\infty}$$
sof doesn't intersect Y

Extrema of the function

In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$\frac{\frac{1}{\sqrt{\log{\left(2 \right)}}} \sqrt{\log{\left(x \right)}}}{2 x \log{\left(x \right)}} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema

Inflection points

Let's find the inflection points, we'll need to solve the equation for this
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$- \frac{2 + \frac{1}{\log{\left(x \right)}}}{4 x^{2} \sqrt{\log{\left(2 \right)}} \sqrt{\log{\left(x \right)}}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = e^{- \frac{1}{2}}$$

Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Have no bends at the whole real axis

Horizontal asymptotes

Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \sqrt{\frac{\log{\left(x \right)}}{\log{\left(2 \right)}}} = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \sqrt{\frac{\log{\left(x \right)}}{\log{\left(2 \right)}}} = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist

Inclined asymptotes

Inclined asymptote can be found by calculating the limit of sqrt(log(x)/log(2)), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\frac{1}{\sqrt{\log{\left(2 \right)}}} \sqrt{\log{\left(x \right)}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\frac{1}{\sqrt{\log{\left(2 \right)}}} \sqrt{\log{\left(x \right)}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left

Even and odd functions

Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:
$$\sqrt{\frac{\log{\left(x \right)}}{\log{\left(2 \right)}}} = \frac{\sqrt{\log{\left(- x \right)}}}{\sqrt{\log{\left(2 \right)}}}$$
- No
$$\sqrt{\frac{\log{\left(x \right)}}{\log{\left(2 \right)}}} = - \frac{\sqrt{\log{\left(- x \right)}}}{\sqrt{\log{\left(2 \right)}}}$$
- No
so, the function
not is
neither even, nor odd

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

Graphing y = sqrt(log2(x))

The graph:

Intersection points:

Piecewise:

The solution