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Plotting a function graph/ log2(cos(2x))

Graphing y = log2(cos(2x))

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       log(cos(2*x))
f(x) = -------------
           log(2)
f(x)=log⁡(cos⁡(2x))log⁡(2)f{\left(x \right)} = \frac{\log{\left(\cos{\left(2 x \right)} \right)}}{\log{\left(2 \right)}} 
f = log(cos(2*x))/log(2)
The graph of the function
02468-8-6-4-2-10105-10
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⁡(cos⁡(2x))log⁡(2)=0\frac{\log{\left(\cos{\left(2 x \right)} \right)}}{\log{\left(2 \right)}} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=πx_{2} = \pi
Numerical solution
x1=94.2477796093519x_{1} = 94.2477796093519
x2=72.2566310277135x_{2} = 72.2566310277135
x3=−97.389372502654x_{3} = -97.389372502654
x4=100.530964736304x_{4} = 100.530964736304
x5=34.5575189914319x_{5} = 34.5575189914319
x6=37.6991120767477x_{6} = 37.6991120767477
x7=56.5486675731909x_{7} = 56.5486675731909
x8=40.8407038692067x_{8} = 40.8407038692067
x9=28.2743338651142x_{9} = 28.2743338651142
x10=−75.3982239198207x_{10} = -75.3982239198207
x11=−43.9822971744191x_{11} = -43.9822971744191
x12=−50.2654822535294x_{12} = -50.2654822535294
x13=18.8495552720944x_{13} = 18.8495552720944
x14=81.681409243074x_{14} = 81.681409243074
x15=6.28318528407908x_{15} = 6.28318528407908
x16=−65.9734457646558x_{16} = -65.9734457646558
x17=43.982297169579x_{17} = 43.982297169579
x18=−37.6991118776023x_{18} = -37.6991118776023
x19=−9.42477817254169x_{19} = -9.42477817254169
x20=−53.4070750099862x_{20} = -53.4070750099862
x21=12.5663707984054x_{21} = 12.5663707984054
x22=−15.7079632968187x_{22} = -15.7079632968187
x23=34.5575194141501x_{23} = 34.5575194141501
x24=−28.274333671219x_{24} = -28.274333671219
x25=−59.6902604582916x_{25} = -59.6902604582916
x26=62.8318524651379x_{26} = 62.8318524651379
x27=0x_{27} = 0
x28=81.6814092224531x_{28} = 81.6814092224531
x29=65.9734457532363x_{29} = 65.9734457532363
x30=−75.3982236054042x_{30} = -75.3982236054042
x31=−94.2477794177114x_{31} = -94.2477794177114
x32=21.9911485852348x_{32} = 21.9911485852348
x33=87.9645943363558x_{33} = 87.9645943363558
x34=−97.3893721997423x_{34} = -97.3893721997423
x35=59.6902606605194x_{35} = 59.6902606605194
x36=−6.28318508874543x_{36} = -6.28318508874543
x37=−72.2566308356894x_{37} = -72.2566308356894
x38=−31.4159267547793x_{38} = -31.4159267547793
x39=59.6902606597981x_{39} = 59.6902606597981
x40=15.7079634939052x_{40} = 15.7079634939052
x41=−31.4159264120844x_{41} = -31.4159264120844
x42=−53.4070753372009x_{42} = -53.4070753372009
x43=−87.9645943581379x_{43} = -87.9645943581379
x44=50.2654824463146x_{44} = 50.2654824463146
x45=12.5663704095248x_{45} = 12.5663704095248
x46=−21.9911485864121x_{46} = -21.9911485864121
x47=−81.6814090389034x_{47} = -81.6814090389034
x48=78.5398161548118x_{48} = 78.5398161548118
x49=84.8230010599183x_{49} = 84.8230010599183
x50=−9.42477780989129x_{50} = -9.42477780989129
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to log(cos(2*x))/log(2).
log⁡(cos⁡(0⋅2))log⁡(2)\frac{\log{\left(\cos{\left(0 \cdot 2 \right)} \right)}}{\log{\left(2 \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
−2sin⁡(2x)log⁡(2)cos⁡(2x)=0- \frac{2 \sin{\left(2 x \right)}}{\log{\left(2 \right)} \cos{\left(2 x \right)}} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=π2x_{2} = \frac{\pi}{2}
The values of the extrema at the points:
(0, 0)

 pi   pi*I  
(--, ------)
 2   log(2)


Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
The function has no minima
Maxima of the function at points:
x2=0x_{2} = 0
Decreasing at intervals
(−∞,0]\left(-\infty, 0\right]
Increasing at intervals
[0,∞)\left[0, \infty\right)
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
−4(sin⁡2(2x)cos⁡2(2x)+1)log⁡(2)=0- \frac{4 \left(\frac{\sin^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}} + 1\right)}{\log{\left(2 \right)}} = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(log⁡(cos⁡(2x))log⁡(2))=log⁡(⟨−1,1⟩)log⁡(2)\lim_{x \to -\infty}\left(\frac{\log{\left(\cos{\left(2 x \right)} \right)}}{\log{\left(2 \right)}}\right) = \frac{\log{\left(\left\langle -1, 1\right\rangle \right)}}{\log{\left(2 \right)}}
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=log⁡(⟨−1,1⟩)log⁡(2)y = \frac{\log{\left(\left\langle -1, 1\right\rangle \right)}}{\log{\left(2 \right)}}
lim⁡x→∞(log⁡(cos⁡(2x))log⁡(2))=log⁡(⟨−1,1⟩)log⁡(2)\lim_{x \to \infty}\left(\frac{\log{\left(\cos{\left(2 x \right)} \right)}}{\log{\left(2 \right)}}\right) = \frac{\log{\left(\left\langle -1, 1\right\rangle \right)}}{\log{\left(2 \right)}}
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=log⁡(⟨−1,1⟩)log⁡(2)y = \frac{\log{\left(\left\langle -1, 1\right\rangle \right)}}{\log{\left(2 \right)}}
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of log(cos(2*x))/log(2), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(log⁡(cos⁡(2x))xlog⁡(2))=0\lim_{x \to -\infty}\left(\frac{\log{\left(\cos{\left(2 x \right)} \right)}}{x \log{\left(2 \right)}}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(log⁡(cos⁡(2x))xlog⁡(2))=0\lim_{x \to \infty}\left(\frac{\log{\left(\cos{\left(2 x \right)} \right)}}{x \log{\left(2 \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⁡(cos⁡(2x))log⁡(2)=log⁡(cos⁡(2x))log⁡(2)\frac{\log{\left(\cos{\left(2 x \right)} \right)}}{\log{\left(2 \right)}} = \frac{\log{\left(\cos{\left(2 x \right)} \right)}}{\log{\left(2 \right)}}
- Yes
log⁡(cos⁡(2x))log⁡(2)=−log⁡(cos⁡(2x))log⁡(2)\frac{\log{\left(\cos{\left(2 x \right)} \right)}}{\log{\left(2 \right)}} = - \frac{\log{\left(\cos{\left(2 x \right)} \right)}}{\log{\left(2 \right)}}
- No
so, the function
is
even
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