Mister Exam
Graphing y = (log(x))/(log(x-3))
The solution
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log(x)
f(x) = ----------
log(x - 3)
$$f{\left(x \right)} = \frac{\log{\left(x \right)}}{\log{\left(x - 3 \right)}}$$
f = log(x)/log(x - 1*3)
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 4$$
$$x_{1} = 4$$
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:
$$\frac{\log{\left(x \right)}}{\log{\left(x - 3 \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 1$$
Numerical solution
$$x_{1} = 1$$
so we need to solve the equation:
$$\frac{\log{\left(x \right)}}{\log{\left(x - 3 \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 1$$
Numerical solution
$$x_{1} = 1$$
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
$$- \frac{\log{\left(x \right)}}{\left(x - 3\right) \log{\left(x - 3 \right)}^{2}} + \frac{1}{x \log{\left(x - 3 \right)}} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$- \frac{\log{\left(x \right)}}{\left(x - 3\right) \log{\left(x - 3 \right)}^{2}} + \frac{1}{x \log{\left(x - 3 \right)}} = 0$$
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
$$\frac{\frac{\left(1 + \frac{2}{\log{\left(x - 3 \right)}}\right) \log{\left(x \right)}}{\left(x - 3\right)^{2} \log{\left(x - 3 \right)}} - \frac{2}{x \left(x - 3\right) \log{\left(x - 3 \right)}} - \frac{1}{x^{2}}}{\log{\left(x - 3 \right)}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 13332.8237448251$$
$$x_{2} = 28428.6795170842$$
$$x_{3} = 19850.0196519154$$
$$x_{4} = 24457.1044346087$$
$$x_{5} = 27102.7423370112$$
$$x_{6} = 15929.0161783672$$
$$x_{7} = 17885.8900670147$$
$$x_{8} = 21820.6302064839$$
$$x_{9} = 10112.8180241462$$
$$x_{10} = 12041.10583426$$
$$x_{11} = 8834.36068835714$$
$$x_{12} = 31086.2572000917$$
$$x_{13} = 10754.2492702545$$
$$x_{14} = 16580.4466160147$$
$$x_{15} = 17232.7496265109$$
$$x_{16} = 29092.3793698505$$
$$x_{17} = 29756.550048717$$
$$x_{18} = 13980.3539259157$$
$$x_{19} = 26440.5315727178$$
$$x_{20} = 15278.4961480115$$
$$x_{21} = 30421.1797514712$$
$$x_{22} = 27765.462852512$$
$$x_{23} = 23797.0832448936$$
$$x_{24} = 19194.5545914475$$
$$x_{25} = 12686.3888212049$$
$$x_{26} = 14628.9273475505$$
$$x_{27} = 25117.6971957888$$
$$x_{28} = 21163.0817373778$$
$$x_{29} = 9472.82499388425$$
$$x_{30} = 20506.2037125583$$
$$x_{31} = 11397.0367067285$$
$$x_{32} = 18539.8352255902$$
$$x_{33} = 23137.6512293001$$
$$x_{34} = 22478.8269890019$$
$$x_{35} = 25778.8448494852$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 4$$
$$\lim_{x \to 4^-}\left(\frac{\frac{\left(1 + \frac{2}{\log{\left(x - 3 \right)}}\right) \log{\left(x \right)}}{\left(x - 3\right)^{2} \log{\left(x - 3 \right)}} - \frac{2}{x \left(x - 3\right) \log{\left(x - 3 \right)}} - \frac{1}{x^{2}}}{\log{\left(x - 3 \right)}}\right) = -\infty$$
Let's take the limit
$$\lim_{x \to 4^+}\left(\frac{\frac{\left(1 + \frac{2}{\log{\left(x - 3 \right)}}\right) \log{\left(x \right)}}{\left(x - 3\right)^{2} \log{\left(x - 3 \right)}} - \frac{2}{x \left(x - 3\right) \log{\left(x - 3 \right)}} - \frac{1}{x^{2}}}{\log{\left(x - 3 \right)}}\right) = \infty$$
Let's take the limit
- the limits are not equal, so
$$x_{1} = 4$$
- is an inflection point
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
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{\frac{\left(1 + \frac{2}{\log{\left(x - 3 \right)}}\right) \log{\left(x \right)}}{\left(x - 3\right)^{2} \log{\left(x - 3 \right)}} - \frac{2}{x \left(x - 3\right) \log{\left(x - 3 \right)}} - \frac{1}{x^{2}}}{\log{\left(x - 3 \right)}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 13332.8237448251$$
$$x_{2} = 28428.6795170842$$
$$x_{3} = 19850.0196519154$$
$$x_{4} = 24457.1044346087$$
$$x_{5} = 27102.7423370112$$
$$x_{6} = 15929.0161783672$$
$$x_{7} = 17885.8900670147$$
$$x_{8} = 21820.6302064839$$
$$x_{9} = 10112.8180241462$$
$$x_{10} = 12041.10583426$$
$$x_{11} = 8834.36068835714$$
$$x_{12} = 31086.2572000917$$
$$x_{13} = 10754.2492702545$$
$$x_{14} = 16580.4466160147$$
$$x_{15} = 17232.7496265109$$
$$x_{16} = 29092.3793698505$$
$$x_{17} = 29756.550048717$$
$$x_{18} = 13980.3539259157$$
$$x_{19} = 26440.5315727178$$
$$x_{20} = 15278.4961480115$$
$$x_{21} = 30421.1797514712$$
$$x_{22} = 27765.462852512$$
$$x_{23} = 23797.0832448936$$
$$x_{24} = 19194.5545914475$$
$$x_{25} = 12686.3888212049$$
$$x_{26} = 14628.9273475505$$
$$x_{27} = 25117.6971957888$$
$$x_{28} = 21163.0817373778$$
$$x_{29} = 9472.82499388425$$
$$x_{30} = 20506.2037125583$$
$$x_{31} = 11397.0367067285$$
$$x_{32} = 18539.8352255902$$
$$x_{33} = 23137.6512293001$$
$$x_{34} = 22478.8269890019$$
$$x_{35} = 25778.8448494852$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 4$$
$$\lim_{x \to 4^-}\left(\frac{\frac{\left(1 + \frac{2}{\log{\left(x - 3 \right)}}\right) \log{\left(x \right)}}{\left(x - 3\right)^{2} \log{\left(x - 3 \right)}} - \frac{2}{x \left(x - 3\right) \log{\left(x - 3 \right)}} - \frac{1}{x^{2}}}{\log{\left(x - 3 \right)}}\right) = -\infty$$
Let's take the limit
$$\lim_{x \to 4^+}\left(\frac{\frac{\left(1 + \frac{2}{\log{\left(x - 3 \right)}}\right) \log{\left(x \right)}}{\left(x - 3\right)^{2} \log{\left(x - 3 \right)}} - \frac{2}{x \left(x - 3\right) \log{\left(x - 3 \right)}} - \frac{1}{x^{2}}}{\log{\left(x - 3 \right)}}\right) = \infty$$
Let's take the limit
- the limits are not equal, so
$$x_{1} = 4$$
- is an inflection point
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
Vertical asymptotes
Have:
$$x_{1} = 4$$
$$x_{1} = 4$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x \right)}}{\log{\left(x - 3 \right)}}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 1$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(x \right)}}{\log{\left(x - 3 \right)}}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 1$$
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x \right)}}{\log{\left(x - 3 \right)}}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 1$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(x \right)}}{\log{\left(x - 3 \right)}}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 1$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of log(x)/log(x - 1*3), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x \right)}}{x \log{\left(x - 3 \right)}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\log{\left(x \right)}}{x \log{\left(x - 3 \right)}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x \right)}}{x \log{\left(x - 3 \right)}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\log{\left(x \right)}}{x \log{\left(x - 3 \right)}}\right) = 0$$
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:
$$\frac{\log{\left(x \right)}}{\log{\left(x - 3 \right)}} = \frac{\log{\left(- x \right)}}{\log{\left(- x - 3 \right)}}$$
- No
$$\frac{\log{\left(x \right)}}{\log{\left(x - 3 \right)}} = - \frac{\log{\left(- x \right)}}{\log{\left(- x - 3 \right)}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\log{\left(x \right)}}{\log{\left(x - 3 \right)}} = \frac{\log{\left(- x \right)}}{\log{\left(- x - 3 \right)}}$$
- No
$$\frac{\log{\left(x \right)}}{\log{\left(x - 3 \right)}} = - \frac{\log{\left(- x \right)}}{\log{\left(- x - 3 \right)}}$$
- No
so, the function
not is
neither even, nor odd