Mister Exam

Other calculators

Plotting a function graph/ x^log4(x-2)

Graphing y = x^log4(x-2)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
        log(x - 2)
        ----------
          log(4)  
f(x) = x
f(x)=xlog⁡(x−2)log⁡(4)f{\left(x \right)} = x^{\frac{\log{\left(x - 2 \right)}}{\log{\left(4 \right)}}} 
f = x^(log(x - 2)/log(4))
The graph of the function
02468-8-6-4-2-1010050
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:
xlog⁡(x−2)log⁡(4)=0x^{\frac{\log{\left(x - 2 \right)}}{\log{\left(4 \right)}}} = 0
Solve this equation
The points of intersection with the axis X:

Numerical solution
x1=2x_{1} = 2
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x^(log(x - 2)/log(4)).
0log⁡(−2)log⁡(4)0^{\frac{\log{\left(-2 \right)}}{\log{\left(4 \right)}}}
The result:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- the solutions of the equation d'not exist
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\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:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
xlog⁡(x−2)log⁡(4)(log⁡(x)(x−2)log⁡(4)+log⁡(x−2)xlog⁡(4))=0x^{\frac{\log{\left(x - 2 \right)}}{\log{\left(4 \right)}}} \left(\frac{\log{\left(x \right)}}{\left(x - 2\right) \log{\left(4 \right)}} + \frac{\log{\left(x - 2 \right)}}{x \log{\left(4 \right)}}\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
d2dx2f(x)=0\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:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
xlog⁡(x−2)log⁡(4)((log⁡(x)x−2+log⁡(x−2)x)2log⁡(4)−log⁡(x)(x−2)2+2x(x−2)−log⁡(x−2)x2)log⁡(4)=0\frac{x^{\frac{\log{\left(x - 2 \right)}}{\log{\left(4 \right)}}} \left(\frac{\left(\frac{\log{\left(x \right)}}{x - 2} + \frac{\log{\left(x - 2 \right)}}{x}\right)^{2}}{\log{\left(4 \right)}} - \frac{\log{\left(x \right)}}{\left(x - 2\right)^{2}} + \frac{2}{x \left(x - 2\right)} - \frac{\log{\left(x - 2 \right)}}{x^{2}}\right)}{\log{\left(4 \right)}} = 0
Solve this equation
The roots of this equation
x1=2.55757872291725x_{1} = 2.55757872291725

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:
Concave at the intervals
[2.55757872291725,∞)\left[2.55757872291725, \infty\right)
Convex at the intervals
(−∞,2.55757872291725]\left(-\infty, 2.55757872291725\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞xlog⁡(x−2)log⁡(4)=∞\lim_{x \to -\infty} x^{\frac{\log{\left(x - 2 \right)}}{\log{\left(4 \right)}}} = \infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞xlog⁡(x−2)log⁡(4)=∞\lim_{x \to \infty} x^{\frac{\log{\left(x - 2 \right)}}{\log{\left(4 \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 x^(log(x - 2)/log(4)), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(xlog⁡(x−2)log⁡(4)x)=0\lim_{x \to -\infty}\left(\frac{x^{\frac{\log{\left(x - 2 \right)}}{\log{\left(4 \right)}}}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(xlog⁡(x−2)log⁡(4)x)=∞\lim_{x \to \infty}\left(\frac{x^{\frac{\log{\left(x - 2 \right)}}{\log{\left(4 \right)}}}}{x}\right) = \infty
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
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:
xlog⁡(x−2)log⁡(4)=(−x)log⁡(−x−2)log⁡(4)x^{\frac{\log{\left(x - 2 \right)}}{\log{\left(4 \right)}}} = \left(- x\right)^{\frac{\log{\left(- x - 2 \right)}}{\log{\left(4 \right)}}}
- No
xlog⁡(x−2)log⁡(4)=−(−x)log⁡(−x−2)log⁡(4)x^{\frac{\log{\left(x - 2 \right)}}{\log{\left(4 \right)}}} = - \left(- x\right)^{\frac{\log{\left(- x - 2 \right)}}{\log{\left(4 \right)}}}
- No
so, the function
not is
neither even, nor odd
    © Mister Exam – Calculator