Mister Exam
Graphing y = log3(x)-log2(x)-1
The solution
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log(x) log(x)
f(x) = ------ - ------ - 1
log(3) log(2)
$$f{\left(x \right)} = \left(- \frac{\log{\left(x \right)}}{\log{\left(2 \right)}} + \frac{\log{\left(x \right)}}{\log{\left(3 \right)}}\right) - 1$$
f = -log(x)/log(2) + log(x)/log(3) - 1
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:
$$\left(- \frac{\log{\left(x \right)}}{\log{\left(2 \right)}} + \frac{\log{\left(x \right)}}{\log{\left(3 \right)}}\right) - 1 = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 3^{\log{\left(2^{\frac{1}{\log{\left(\frac{2}{3} \right)}}} \right)}}$$
Numerical solution
$$x_{1} = 0.152881814200196$$
so we need to solve the equation:
$$\left(- \frac{\log{\left(x \right)}}{\log{\left(2 \right)}} + \frac{\log{\left(x \right)}}{\log{\left(3 \right)}}\right) - 1 = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 3^{\log{\left(2^{\frac{1}{\log{\left(\frac{2}{3} \right)}}} \right)}}$$
Numerical solution
$$x_{1} = 0.152881814200196$$
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
$$- \frac{1}{x \log{\left(2 \right)}} + \frac{1}{x \log{\left(3 \right)}} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$- \frac{1}{x \log{\left(2 \right)}} + \frac{1}{x \log{\left(3 \right)}} = 0$$
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
$$\frac{- \frac{1}{\log{\left(3 \right)}} + \frac{1}{\log{\left(2 \right)}}}{x^{2}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{- \frac{1}{\log{\left(3 \right)}} + \frac{1}{\log{\left(2 \right)}}}{x^{2}} = 0$$
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 x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\left(- \frac{\log{\left(x \right)}}{\log{\left(2 \right)}} + \frac{\log{\left(x \right)}}{\log{\left(3 \right)}}\right) - 1\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\left(- \frac{\log{\left(x \right)}}{\log{\left(2 \right)}} + \frac{\log{\left(x \right)}}{\log{\left(3 \right)}}\right) - 1\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\left(- \frac{\log{\left(x \right)}}{\log{\left(2 \right)}} + \frac{\log{\left(x \right)}}{\log{\left(3 \right)}}\right) - 1\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\left(- \frac{\log{\left(x \right)}}{\log{\left(2 \right)}} + \frac{\log{\left(x \right)}}{\log{\left(3 \right)}}\right) - 1\right) = -\infty$$
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(x)/log(3) - log(x)/log(2) - 1, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left(- \frac{\log{\left(x \right)}}{\log{\left(2 \right)}} + \frac{\log{\left(x \right)}}{\log{\left(3 \right)}}\right) - 1}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\left(- \frac{\log{\left(x \right)}}{\log{\left(2 \right)}} + \frac{\log{\left(x \right)}}{\log{\left(3 \right)}}\right) - 1}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\left(- \frac{\log{\left(x \right)}}{\log{\left(2 \right)}} + \frac{\log{\left(x \right)}}{\log{\left(3 \right)}}\right) - 1}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\left(- \frac{\log{\left(x \right)}}{\log{\left(2 \right)}} + \frac{\log{\left(x \right)}}{\log{\left(3 \right)}}\right) - 1}{x}\right) = 0$$
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:
$$\left(- \frac{\log{\left(x \right)}}{\log{\left(2 \right)}} + \frac{\log{\left(x \right)}}{\log{\left(3 \right)}}\right) - 1 = - \frac{\log{\left(- x \right)}}{\log{\left(2 \right)}} + \frac{\log{\left(- x \right)}}{\log{\left(3 \right)}} - 1$$
- No
$$\left(- \frac{\log{\left(x \right)}}{\log{\left(2 \right)}} + \frac{\log{\left(x \right)}}{\log{\left(3 \right)}}\right) - 1 = - \frac{\log{\left(- x \right)}}{\log{\left(3 \right)}} + \frac{\log{\left(- x \right)}}{\log{\left(2 \right)}} + 1$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\left(- \frac{\log{\left(x \right)}}{\log{\left(2 \right)}} + \frac{\log{\left(x \right)}}{\log{\left(3 \right)}}\right) - 1 = - \frac{\log{\left(- x \right)}}{\log{\left(2 \right)}} + \frac{\log{\left(- x \right)}}{\log{\left(3 \right)}} - 1$$
- No
$$\left(- \frac{\log{\left(x \right)}}{\log{\left(2 \right)}} + \frac{\log{\left(x \right)}}{\log{\left(3 \right)}}\right) - 1 = - \frac{\log{\left(- x \right)}}{\log{\left(3 \right)}} + \frac{\log{\left(- x \right)}}{\log{\left(2 \right)}} + 1$$
- No
so, the function
not is
neither even, nor odd