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Plotting a function graph/ log(y)

Graphing y = log(y)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(y) = log(y)
f(y)=log⁡(y)f{\left(y \right)} = \log{\left(y \right)} 
f = log(y)
The graph of the function
02468-8-6-4-2-10105-5
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis Y at f = 0
so we need to solve the equation:
log⁡(y)=0\log{\left(y \right)} = 0
Solve this equation
The points of intersection with the axis Y:

Analytical solution
y1=1y_{1} = 1
Numerical solution
y1=1y_{1} = 1
The points of intersection with the Y axis coordinate
The graph crosses Y axis when y equals 0:
substitute y = 0 to log(y).
log⁡(0)\log{\left(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
ddyf(y)=0\frac{d}{d y} f{\left(y \right)} = 0
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
ddyf(y)=\frac{d}{d y} f{\left(y \right)} =
the first derivative
1y=0\frac{1}{y} = 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
d2dy2f(y)=0\frac{d^{2}}{d y^{2}} f{\left(y \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dy2f(y)=\frac{d^{2}}{d y^{2}} f{\left(y \right)} =
the second derivative
−1y2=0- \frac{1}{y^{2}} = 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 y->+oo and y->-oo
lim⁡y→−∞log⁡(y)=∞\lim_{y \to -\infty} \log{\left(y \right)} = \infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡y→∞log⁡(y)=∞\lim_{y \to \infty} \log{\left(y \right)} = \infty
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of log(y), divided by y at y->+oo and y ->-oo
lim⁡y→−∞(log⁡(y)y)=0\lim_{y \to -\infty}\left(\frac{\log{\left(y \right)}}{y}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡y→∞(log⁡(y)y)=0\lim_{y \to \infty}\left(\frac{\log{\left(y \right)}}{y}\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(-y) и f = -f(-y).
So, check:
log⁡(y)=log⁡(−y)\log{\left(y \right)} = \log{\left(- y \right)}
- No
log⁡(y)=−log⁡(−y)\log{\left(y \right)} = - \log{\left(- y \right)}
- No
so, the function
not is
neither even, nor odd
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