Graphing y = lnt

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f(t) = log(t)

f{\left(t \right)} = \log{\left(t \right)}

f = log(t)

The graph of the function

The points of intersection with the X-axis coordinate

Graph of the function intersects the axis T at f = 0
so we need to solve the equation:

\log{\left(t \right)} = 0

Solve this equation
The points of intersection with the axis T:

Analytical solution

t_{1} = 1

Numerical solution

t_{1} = 1

The points of intersection with the Y axis coordinate

The graph crosses Y axis when t equals 0:
substitute t = 0 to log(t).

\log{\left(0 \right)}

The result:

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

\frac{d}{d t} f{\left(t \right)} = 0

(the derivative equals zero),
and the roots of this equation are the extrema of this function:

\frac{d}{d t} f{\left(t \right)} =

\frac{1}{t} = 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

\frac{d^{2}}{d t^{2}} f{\left(t \right)} = 0

(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:

\frac{d^{2}}{d t^{2}} f{\left(t \right)} =

- \frac{1}{t^{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 t->+oo and t->-oo

\lim_{t \to -\infty} \log{\left(t \right)} = \infty

Let's take the limit
so,
horizontal asymptote on the left doesn’t exist

\lim_{t \to \infty} \log{\left(t \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(t), divided by t at t->+oo and t ->-oo

\lim_{t \to -\infty}\left(\frac{\log{\left(t \right)}}{t}\right) = 0

Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right

\lim_{t \to \infty}\left(\frac{\log{\left(t \right)}}{t}\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(-t) и f = -f(-t).
So, check:

\log{\left(t \right)} = \log{\left(- t \right)}

- No

\log{\left(t \right)} = - \log{\left(- t \right)}

- No
so, the function
not is
neither even, nor odd