Mister Exam

# Graphing y = y=2log2(x-1)

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### The solution

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2*log(x - 1)
f(x) = ------------
log(2)
$$f{\left(x \right)} = \frac{2 \log{\left(x - 1 \right)}}{\log{\left(2 \right)}}$$
f = 2*log(x - 1*1)/log(2)
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:
$$\frac{2 \log{\left(x - 1 \right)}}{\log{\left(2 \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:

Analytical solution
$$x_{1} = 2$$
Numerical solution
$$x_{1} = 2$$
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 2*log(x - 1*1)/log(2).
$$\frac{2 \log{\left(\left(-1\right) 1 + 0 \right)}}{\log{\left(2 \right)}}$$
The result:
$$f{\left(0 \right)} = \frac{2 i \pi}{\log{\left(2 \right)}}$$
The point:
(0, 2*pi*i/log(2))
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
$$\frac{2}{\left(x - 1\right) \log{\left(2 \right)}} = 0$$
Solve this equation
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
$$- \frac{2}{\left(x - 1\right)^{2} \log{\left(2 \right)}} = 0$$
Solve this equation
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
$$\lim_{x \to -\infty}\left(\frac{2 \log{\left(x - 1 \right)}}{\log{\left(2 \right)}}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{2 \log{\left(x - 1 \right)}}{\log{\left(2 \right)}}\right) = \infty$$
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 2*log(x - 1*1)/log(2), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{2 \log{\left(x - 1 \right)}}{x \log{\left(2 \right)}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{2 \log{\left(x - 1 \right)}}{x \log{\left(2 \right)}}\right) = 0$$
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:
$$\frac{2 \log{\left(x - 1 \right)}}{\log{\left(2 \right)}} = \frac{2 \log{\left(- x - 1 \right)}}{\log{\left(2 \right)}}$$
- No
$$\frac{2 \log{\left(x - 1 \right)}}{\log{\left(2 \right)}} = - \frac{2 \log{\left(- x - 1 \right)}}{\log{\left(2 \right)}}$$
- No
so, the function
not is
neither even, nor odd