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Plotting a function graph/ ln(cos(x))

Graphing y = ln(cos(x))

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = log(cos(x))
f(x)=log⁡(cos⁡(x))f{\left(x \right)} = \log{\left(\cos{\left(x \right)} \right)} 
f = log(cos(x))
The graph of the function
0.000.050.100.150.200.250.300.350.400.450.500.550.600.650.700.750.5-0.5
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⁡(x))=0\log{\left(\cos{\left(x \right)} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=2πx_{2} = 2 \pi
Numerical solution
x1=25.1327409700176x_{1} = 25.1327409700176
x2=50.2654824463311x_{2} = 50.2654824463311
x3=56.5486679766099x_{3} = 56.5486679766099
x4=−69.1150374752626x_{4} = -69.1150374752626
x5=−25.1327403562086x_{5} = -25.1327403562086
x6=87.9645942296464x_{6} = 87.9645942296464
x7=−75.3982234018825x_{7} = -75.3982234018825
x8=−87.9645943355219x_{8} = -87.9645943355219
x9=−37.6991118773736x_{9} = -37.6991118773736
x10=−12.5663702522378x_{10} = -12.5663702522378
x11=−62.8318534517187x_{11} = -62.8318534517187
x12=56.5486675932357x_{12} = 56.5486675932357
x13=50.2654824640562x_{13} = 50.2654824640562
x14=25.1327418930934x_{14} = 25.1327418930934
x15=25.1327406563971x_{15} = 25.1327406563971
x16=6.28318532165763x_{16} = 6.28318532165763
x17=94.2477796068599x_{17} = 94.2477796068599
x18=69.1150378238503x_{18} = 69.1150378238503
x19=31.4159255304025x_{19} = 31.4159255304025
x20=87.9645943360512x_{20} = 87.9645943360512
x21=−6.28318566745615x_{21} = -6.28318566745615
x22=−56.5486687640637x_{22} = -56.5486687640637
x23=69.1150390127643x_{23} = 69.1150390127643
x24=6.28318528416623x_{24} = 6.28318528416623
x25=−75.3982238864105x_{25} = -75.3982238864105
x26=−25.1327415878584x_{26} = -25.1327415878584
x27=75.3982227418079x_{27} = 75.3982227418079
x28=31.4159269101267x_{28} = 31.4159269101267
x29=−100.53096457631x_{29} = -100.53096457631
x30=18.8495555741382x_{30} = 18.8495555741382
x31=43.982297089421x_{31} = 43.982297089421
x32=−18.8495562408585x_{32} = -18.8495562408585
x33=−62.8318536803612x_{33} = -62.8318536803612
x34=−12.5663716386669x_{34} = -12.5663716386669
x35=43.9822971695019x_{35} = 43.9822971695019
x36=0x_{36} = 0
x37=31.4159255531763x_{37} = 31.4159255531763
x38=−69.1150387500801x_{38} = -69.1150387500801
x39=81.6814085526449x_{39} = 81.6814085526449
x40=−6.28318511692891x_{40} = -6.28318511692891
x41=69.1150390932802x_{41} = 69.1150390932802
x42=−18.8495565116576x_{42} = -18.8495565116576
x43=−18.8495553258088x_{43} = -18.8495553258088
x44=−69.1150373853363x_{44} = -69.1150373853363
x45=62.831852735923x_{45} = 62.831852735923
x46=−87.9645943584596x_{46} = -87.9645943584596
x47=−62.8318528736237x_{47} = -62.8318528736237
x48=100.530965106382x_{48} = 100.530965106382
x49=−43.9822971932261x_{49} = -43.9822971932261
x50=75.39822407273x_{50} = 75.39822407273
x51=100.530964753022x_{51} = 100.530964753022
x52=12.5663704334084x_{52} = 12.5663704334084
x53=−12.5663716213936x_{53} = -12.5663716213936
x54=−31.4159262776781x_{54} = -31.4159262776781
x55=37.6991114441887x_{55} = 37.6991114441887
x56=−94.2477799001796x_{56} = -94.2477799001796
x57=−100.530965897751x_{57} = -100.530965897751
x58=−50.2654822771894x_{58} = -50.2654822771894
x59=−50.2654827822791x_{59} = -50.2654827822791
x60=81.681409203672x_{60} = 81.681409203672
x61=25.1327418431203x_{61} = 25.1327418431203
x62=−62.8318524940769x_{62} = -62.8318524940769
x63=62.8318542034359x_{63} = 62.8318542034359
x64=−81.6814090384469x_{64} = -81.6814090384469
x65=69.1150381807919x_{65} = 69.1150381807919
x66=−56.5486674143785x_{66} = -56.5486674143785
x67=12.5663708485373x_{67} = 12.5663708485373
x68=62.8318538684035x_{68} = 62.8318538684035
x69=−18.8495557286473x_{69} = -18.8495557286473
x70=−31.4159267264704x_{70} = -31.4159267264704
x71=−56.5486688343165x_{71} = -56.5486688343165
x72=18.8495570029843x_{72} = 18.8495570029843
x73=−37.6991118203008x_{73} = -37.6991118203008
x74=−94.2477794374461x_{74} = -94.2477794374461
x75=94.2477796093522x_{75} = 94.2477796093522
x76=18.8495567580196x_{76} = 18.8495567580196
x77=75.3982226911418x_{77} = 75.3982226911418
x78=−81.6814089617871x_{78} = -81.6814089617871
x79=−25.1327401930409x_{79} = -25.1327401930409
x80=−43.9822971744998x_{80} = -43.9822971744998
x81=37.6991120433529x_{81} = 37.6991120433529
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to log(cos(x)).
log⁡(cos⁡(0))\log{\left(\cos{\left(0 \right)} \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
−sin⁡(x)cos⁡(x)=0- \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=πx_{2} = \pi
The values of the extrema at the points:
(0, 0)

(pi, pi*I)


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
−(sin⁡2(x)cos⁡2(x)+1)=0- (\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1) = 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⁡(x))=log⁡(⟨−1,1⟩)\lim_{x \to -\infty} \log{\left(\cos{\left(x \right)} \right)} = \log{\left(\left\langle -1, 1\right\rangle \right)}
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=log⁡(⟨−1,1⟩)y = \log{\left(\left\langle -1, 1\right\rangle \right)}
lim⁡x→∞log⁡(cos⁡(x))=log⁡(⟨−1,1⟩)\lim_{x \to \infty} \log{\left(\cos{\left(x \right)} \right)} = \log{\left(\left\langle -1, 1\right\rangle \right)}
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=log⁡(⟨−1,1⟩)y = \log{\left(\left\langle -1, 1\right\rangle \right)}
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of log(cos(x)), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(log⁡(cos⁡(x))x)=0\lim_{x \to -\infty}\left(\frac{\log{\left(\cos{\left(x \right)} \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(log⁡(cos⁡(x))x)=0\lim_{x \to \infty}\left(\frac{\log{\left(\cos{\left(x \right)} \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:
log⁡(cos⁡(x))=log⁡(cos⁡(x))\log{\left(\cos{\left(x \right)} \right)} = \log{\left(\cos{\left(x \right)} \right)}
- Yes
log⁡(cos⁡(x))=−log⁡(cos⁡(x))\log{\left(\cos{\left(x \right)} \right)} = - \log{\left(\cos{\left(x \right)} \right)}
- No
so, the function
is
even
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