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Plotting a function graph/ log(x+1)/log(x-1)

Graphing y = log(x+1)/log(x-1)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       log(x + 1)
f(x) = ----------
       log(x - 1)
f(x)=log⁡(x+1)log⁡(x−1)f{\left(x \right)} = \frac{\log{\left(x + 1 \right)}}{\log{\left(x - 1 \right)}} 
f = log(x + 1)/log(x - 1)
The graph of the function
-1.0-0.53.00.00.51.01.52.02.5-2000020000
The domain of the function
The points at which the function is not precisely defined:
x1=2x_{1} = 2
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+1)log⁡(x−1)=0\frac{\log{\left(x + 1 \right)}}{\log{\left(x - 1 \right)}} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=0x_{1} = 0
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to log(x + 1)/log(x - 1).
log⁡(1)log⁡(−1)\frac{\log{\left(1 \right)}}{\log{\left(-1 \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
1(x+1)log⁡(x−1)−log⁡(x+1)(x−1)log⁡(x−1)2=0\frac{1}{\left(x + 1\right) \log{\left(x - 1 \right)}} - \frac{\log{\left(x + 1 \right)}}{\left(x - 1\right) \log{\left(x - 1 \right)}^{2}} = 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−1))log⁡(x+1)(x−1)2log⁡(x−1)−1(x+1)2−2(x−1)(x+1)log⁡(x−1)log⁡(x−1)=0\frac{\frac{\left(1 + \frac{2}{\log{\left(x - 1 \right)}}\right) \log{\left(x + 1 \right)}}{\left(x - 1\right)^{2} \log{\left(x - 1 \right)}} - \frac{1}{\left(x + 1\right)^{2}} - \frac{2}{\left(x - 1\right) \left(x + 1\right) \log{\left(x - 1 \right)}}}{\log{\left(x - 1 \right)}} = 0
Solve this equation
The roots of this equation
x1=24990.9819022814x_{1} = 24990.9819022814
x2=23012.180065108x_{2} = 23012.180065108
x3=21696.0720321665x_{3} = 21696.0720321665
x4=11927.1424610773x_{4} = 11927.1424610773
x5=13216.8567271946x_{5} = 13216.8567271946
x6=26974.9097289087x_{6} = 26974.9097289087
x7=27637.2822298x_{7} = 27637.2822298
x8=18417.9306531122x_{8} = 18417.9306531122
x9=19726.98181305x_{9} = 19726.98181305
x10=15159.9785186663x_{10} = 15159.9785186663
x11=31621.7160682476x_{11} = 31621.7160682476
x12=17112.093643241x_{12} = 17112.093643241
x13=30956.4913579122x_{13} = 30956.4913579122
x14=19072.0701365837x_{14} = 19072.0701365837
x15=16460.4647533393x_{15} = 16460.4647533393
x16=30291.7120950935x_{16} = 30291.7120950935
x17=8091.91985555771x_{17} = 8091.91985555771
x18=12571.3885228224x_{18} = 12571.3885228224
x19=10642.6085702957x_{19} = 10642.6085702957
x20=25651.7441696976x_{20} = 25651.7441696976
x21=7458.74655589038x_{21} = 7458.74655589038
x22=10002.4869770251x_{22} = 10002.4869770251
x23=14511.2077796967x_{23} = 14511.2077796967
x24=11284.1895805989x_{24} = 11284.1895805989
x25=8727.02556497286x_{25} = 8727.02556497286
x26=23671.1817540481x_{26} = 23671.1817540481
x27=28300.161960649x_{27} = 28300.161960649
x28=28963.5353600946x_{28} = 28963.5353600946
x29=29627.3895105595x_{29} = 29627.3895105595
x30=22353.8031957521x_{30} = 22353.8031957521
x31=24330.7885666891x_{31} = 24330.7885666891
x32=13863.4826786086x_{32} = 13863.4826786086
x33=21039.0087570418x_{33} = 21039.0087570418
x34=15809.7458648551x_{34} = 15809.7458648551
x35=20382.6369684156x_{35} = 20382.6369684156
x36=9363.92266180088x_{36} = 9363.92266180088
x37=17764.594137361x_{37} = 17764.594137361
x38=26313.0587104048x_{38} = 26313.0587104048
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=2x_{1} = 2

lim⁡x→2−((1+2log⁡(x−1))log⁡(x+1)(x−1)2log⁡(x−1)−1(x+1)2−2(x−1)(x+1)log⁡(x−1)log⁡(x−1))=−∞\lim_{x \to 2^-}\left(\frac{\frac{\left(1 + \frac{2}{\log{\left(x - 1 \right)}}\right) \log{\left(x + 1 \right)}}{\left(x - 1\right)^{2} \log{\left(x - 1 \right)}} - \frac{1}{\left(x + 1\right)^{2}} - \frac{2}{\left(x - 1\right) \left(x + 1\right) \log{\left(x - 1 \right)}}}{\log{\left(x - 1 \right)}}\right) = -\infty
lim⁡x→2+((1+2log⁡(x−1))log⁡(x+1)(x−1)2log⁡(x−1)−1(x+1)2−2(x−1)(x+1)log⁡(x−1)log⁡(x−1))=∞\lim_{x \to 2^+}\left(\frac{\frac{\left(1 + \frac{2}{\log{\left(x - 1 \right)}}\right) \log{\left(x + 1 \right)}}{\left(x - 1\right)^{2} \log{\left(x - 1 \right)}} - \frac{1}{\left(x + 1\right)^{2}} - \frac{2}{\left(x - 1\right) \left(x + 1\right) \log{\left(x - 1 \right)}}}{\log{\left(x - 1 \right)}}\right) = \infty
- the limits are not equal, so
x1=2x_{1} = 2
- 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=2x_{1} = 2
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+1)log⁡(x−1))=1\lim_{x \to -\infty}\left(\frac{\log{\left(x + 1 \right)}}{\log{\left(x - 1 \right)}}\right) = 1
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=1y = 1
lim⁡x→∞(log⁡(x+1)log⁡(x−1))=1\lim_{x \to \infty}\left(\frac{\log{\left(x + 1 \right)}}{\log{\left(x - 1 \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 + 1)/log(x - 1), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(log⁡(x+1)xlog⁡(x−1))=0\lim_{x \to -\infty}\left(\frac{\log{\left(x + 1 \right)}}{x \log{\left(x - 1 \right)}}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(log⁡(x+1)xlog⁡(x−1))=0\lim_{x \to \infty}\left(\frac{\log{\left(x + 1 \right)}}{x \log{\left(x - 1 \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+1)log⁡(x−1)=log⁡(1−x)log⁡(−x−1)\frac{\log{\left(x + 1 \right)}}{\log{\left(x - 1 \right)}} = \frac{\log{\left(1 - x \right)}}{\log{\left(- x - 1 \right)}}
- No
log⁡(x+1)log⁡(x−1)=−log⁡(1−x)log⁡(−x−1)\frac{\log{\left(x + 1 \right)}}{\log{\left(x - 1 \right)}} = - \frac{\log{\left(1 - x \right)}}{\log{\left(- x - 1 \right)}}
- No
so, the function
not is
neither even, nor odd
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