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Plotting a function graph/ 1/(ln(x)/(x-3))

Graphing y = 1/(ln(x)/(x-3))

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          1    
f(x) = --------
       /log(x)\
       |------|
       \x - 3 /
f(x)=11x−3log⁡(x)f{\left(x \right)} = \frac{1}{\frac{1}{x - 3} \log{\left(x \right)}} 
f = 1/(log(x)/(x - 3))
The graph of the function
02468-8-6-4-2-1010-100100
The domain of the function
The points at which the function is not precisely defined:
x1=1x_{1} = 1
x2=3x_{2} = 3
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:
11x−3log⁡(x)=0\frac{1}{\frac{1}{x - 3} \log{\left(x \right)}} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=3x_{1} = 3
Numerical solution
x1=3x_{1} = 3
x2=0x_{2} = 0
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 1/(log(x)/(x - 3)).
11−3log⁡(0)\frac{1}{\frac{1}{-3} \log{\left(0 \right)}}
The result:
f(0)=0f{\left(0 \right)} = 0
The point:
(0, 0)
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
x−3log⁡(x)(x−3)(log⁡(x)(x−3)2−1x(x−3))log⁡(x)=0\frac{\frac{x - 3}{\log{\left(x \right)}} \left(x - 3\right) \left(\frac{\log{\left(x \right)}}{\left(x - 3\right)^{2}} - \frac{1}{x \left(x - 3\right)}\right)}{\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
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
(1−x−3xlog⁡(x))(log⁡(x)x−3−1x)+(x−3)(−2log⁡(x)(x−3)2+2x(x−3)+1x2)+log⁡(x)x−3−(x−3)(log⁡(x)x−3−1x)xlog⁡(x)−1xlog⁡(x)2=0\frac{\left(1 - \frac{x - 3}{x \log{\left(x \right)}}\right) \left(\frac{\log{\left(x \right)}}{x - 3} - \frac{1}{x}\right) + \left(x - 3\right) \left(- \frac{2 \log{\left(x \right)}}{\left(x - 3\right)^{2}} + \frac{2}{x \left(x - 3\right)} + \frac{1}{x^{2}}\right) + \frac{\log{\left(x \right)}}{x - 3} - \frac{\left(x - 3\right) \left(\frac{\log{\left(x \right)}}{x - 3} - \frac{1}{x}\right)}{x \log{\left(x \right)}} - \frac{1}{x}}{\log{\left(x \right)}^{2}} = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
x1=1x_{1} = 1
x2=3x_{2} = 3
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞11x−3log⁡(x)=−∞\lim_{x \to -\infty} \frac{1}{\frac{1}{x - 3} \log{\left(x \right)}} = -\infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞11x−3log⁡(x)=∞\lim_{x \to \infty} \frac{1}{\frac{1}{x - 3} \log{\left(x \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 1/(log(x)/(x - 3)), divided by x at x->+oo and x ->-oo
lim⁡x→−∞((x−3)1log⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\left(x - 3\right) \frac{1}{\log{\left(x \right)}}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞((x−3)1log⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\left(x - 3\right) \frac{1}{\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:
11x−3log⁡(x)=−x−3log⁡(−x)\frac{1}{\frac{1}{x - 3} \log{\left(x \right)}} = \frac{- x - 3}{\log{\left(- x \right)}}
- No
11x−3log⁡(x)=−−x−3log⁡(−x)\frac{1}{\frac{1}{x - 3} \log{\left(x \right)}} = - \frac{- x - 3}{\log{\left(- x \right)}}
- No
so, the function
not is
neither even, nor odd
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