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Plotting a function graph/ (log(x))/(log(x-3))

Graphing y = (log(x))/(log(x-3))

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
         log(x)  
f(x) = ----------
       log(x - 3)
f(x)=log⁡(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:
x1=4x_{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:
log⁡(x)log⁡(x−3)=0\frac{\log{\left(x \right)}}{\log{\left(x - 3 \right)}} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=1x_{1} = 1
Numerical solution
x1=1x_{1} = 1
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to log(x)/log(x - 1*3).
log⁡(0)log⁡((−1)3+0)\frac{\log{\left(0 \right)}}{\log{\left(\left(-1\right) 3 + 0 \right)}}
The result:
f(0)=∞~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
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
−log⁡(x)(x−3)log⁡(x−3)2+1xlog⁡(x−3)=0- \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
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+2log⁡(x−3))log⁡(x)(x−3)2log⁡(x−3)−2x(x−3)log⁡(x−3)−1x2log⁡(x−3)=0\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
x1=13332.8237448251x_{1} = 13332.8237448251
x2=28428.6795170842x_{2} = 28428.6795170842
x3=19850.0196519154x_{3} = 19850.0196519154
x4=24457.1044346087x_{4} = 24457.1044346087
x5=27102.7423370112x_{5} = 27102.7423370112
x6=15929.0161783672x_{6} = 15929.0161783672
x7=17885.8900670147x_{7} = 17885.8900670147
x8=21820.6302064839x_{8} = 21820.6302064839
x9=10112.8180241462x_{9} = 10112.8180241462
x10=12041.10583426x_{10} = 12041.10583426
x11=8834.36068835714x_{11} = 8834.36068835714
x12=31086.2572000917x_{12} = 31086.2572000917
x13=10754.2492702545x_{13} = 10754.2492702545
x14=16580.4466160147x_{14} = 16580.4466160147
x15=17232.7496265109x_{15} = 17232.7496265109
x16=29092.3793698505x_{16} = 29092.3793698505
x17=29756.550048717x_{17} = 29756.550048717
x18=13980.3539259157x_{18} = 13980.3539259157
x19=26440.5315727178x_{19} = 26440.5315727178
x20=15278.4961480115x_{20} = 15278.4961480115
x21=30421.1797514712x_{21} = 30421.1797514712
x22=27765.462852512x_{22} = 27765.462852512
x23=23797.0832448936x_{23} = 23797.0832448936
x24=19194.5545914475x_{24} = 19194.5545914475
x25=12686.3888212049x_{25} = 12686.3888212049
x26=14628.9273475505x_{26} = 14628.9273475505
x27=25117.6971957888x_{27} = 25117.6971957888
x28=21163.0817373778x_{28} = 21163.0817373778
x29=9472.82499388425x_{29} = 9472.82499388425
x30=20506.2037125583x_{30} = 20506.2037125583
x31=11397.0367067285x_{31} = 11397.0367067285
x32=18539.8352255902x_{32} = 18539.8352255902
x33=23137.6512293001x_{33} = 23137.6512293001
x34=22478.8269890019x_{34} = 22478.8269890019
x35=25778.8448494852x_{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:
x1=4x_{1} = 4

lim⁡x→4−((1+2log⁡(x−3))log⁡(x)(x−3)2log⁡(x−3)−2x(x−3)log⁡(x−3)−1x2log⁡(x−3))=−∞\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→4+((1+2log⁡(x−3))log⁡(x)(x−3)2log⁡(x−3)−2x(x−3)log⁡(x−3)−1x2log⁡(x−3))=∞\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
x1=4x_{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:
x1=4x_{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→−∞(log⁡(x)log⁡(x−3))=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=1y = 1
lim⁡x→∞(log⁡(x)log⁡(x−3))=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=1y = 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→−∞(log⁡(x)xlog⁡(x−3))=0\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→∞(log⁡(x)xlog⁡(x−3))=0\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:
log⁡(x)log⁡(x−3)=log⁡(−x)log⁡(−x−3)\frac{\log{\left(x \right)}}{\log{\left(x - 3 \right)}} = \frac{\log{\left(- x \right)}}{\log{\left(- x - 3 \right)}}
- No
log⁡(x)log⁡(x−3)=−log⁡(−x)log⁡(−x−3)\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
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