Graphing y = log(1/8)x

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f(x) = log(1/8)*x

$$f{\left(x \right)} = x \log{\left(\frac{1}{8} \right)}$$

f = x*log(1/8)

The graph of the function

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:
$$x \log{\left(\frac{1}{8} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:

Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = 0$$

The points of intersection with the Y axis coordinate

The graph crosses Y axis when x equals 0:
substitute x = 0 to log(1/8)*x.
$$0 \log{\left(\frac{1}{8} \right)}$$
The result:
$$f{\left(0 \right)} = 0$$
The point:

(0, 0)

Extrema of the function

In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$\log{\left(\frac{1}{8} \right)} = 0$$
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
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$0 = 0$$
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
$$\lim_{x \to -\infty}\left(x \log{\left(\frac{1}{8} \right)}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(x \log{\left(\frac{1}{8} \right)}\right) = -\infty$$
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(1/8)*x, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty} \log{\left(\frac{1}{8} \right)} = - \log{\left(8 \right)}$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = - x \log{\left(8 \right)}$$
$$\lim_{x \to \infty} \log{\left(\frac{1}{8} \right)} = - \log{\left(8 \right)}$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = - x \log{\left(8 \right)}$$

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:
$$x \log{\left(\frac{1}{8} \right)} = - x \log{\left(\frac{1}{8} \right)}$$
- No
$$x \log{\left(\frac{1}{8} \right)} = x \log{\left(\frac{1}{8} \right)}$$
- No
so, the function
not is
neither even, nor odd

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Identical expressions

Graphing y = log(1/8)x

The graph:

Intersection points:

Piecewise:

The solution