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3log2(x+5)-4
Plotting a function graph/ 3log2(x+5)-4

Graphing y = 3log2(x+5)-4

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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       3*log(x + 5)    
f(x) = ------------ - 4
          log(2)
f(x)=3log⁡(x+5)log⁡(2)−4f{\left(x \right)} = \frac{3 \log{\left(x + 5 \right)}}{\log{\left(2 \right)}} - 4 
f = 3*log(x + 5)/log(2) - 1*4
The graph of the function
02468-8-6-4-2-1010-5050
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:
3log⁡(x+5)log⁡(2)−4=0\frac{3 \log{\left(x + 5 \right)}}{\log{\left(2 \right)}} - 4 = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−5+2⋅23x_{1} = -5 + 2 \cdot \sqrt[3]{2}
Numerical solution
x1=−2.48015790021025x_{1} = -2.48015790021025
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 3*log(x + 5)/log(2) - 1*4.
(−1)4+3log⁡(0+5)log⁡(2)\left(-1\right) 4 + \frac{3 \log{\left(0 + 5 \right)}}{\log{\left(2 \right)}}
The result:
f(0)=−4+3log⁡(5)log⁡(2)f{\left(0 \right)} = -4 + \frac{3 \log{\left(5 \right)}}{\log{\left(2 \right)}}
The point:
(0, -4 + 3*log(5)/log(2))
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\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:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
3(x+5)log⁡(2)=0\frac{3}{\left(x + 5\right) \log{\left(2 \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
d2dx2f(x)=0\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:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
−3(x+5)2log⁡(2)=0- \frac{3}{\left(x + 5\right)^{2} \log{\left(2 \right)}} = 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→−∞(3log⁡(x+5)log⁡(2)−4)=∞\lim_{x \to -\infty}\left(\frac{3 \log{\left(x + 5 \right)}}{\log{\left(2 \right)}} - 4\right) = \infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞(3log⁡(x+5)log⁡(2)−4)=∞\lim_{x \to \infty}\left(\frac{3 \log{\left(x + 5 \right)}}{\log{\left(2 \right)}} - 4\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 3*log(x + 5)/log(2) - 1*4, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(3log⁡(x+5)log⁡(2)−4x)=0\lim_{x \to -\infty}\left(\frac{\frac{3 \log{\left(x + 5 \right)}}{\log{\left(2 \right)}} - 4}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(3log⁡(x+5)log⁡(2)−4x)=0\lim_{x \to \infty}\left(\frac{\frac{3 \log{\left(x + 5 \right)}}{\log{\left(2 \right)}} - 4}{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:
3log⁡(x+5)log⁡(2)−4=3log⁡(−x+5)log⁡(2)−4\frac{3 \log{\left(x + 5 \right)}}{\log{\left(2 \right)}} - 4 = \frac{3 \log{\left(- x + 5 \right)}}{\log{\left(2 \right)}} - 4
- No
3log⁡(x+5)log⁡(2)−4=−3log⁡(−x+5)log⁡(2)+4\frac{3 \log{\left(x + 5 \right)}}{\log{\left(2 \right)}} - 4 = - \frac{3 \log{\left(- x + 5 \right)}}{\log{\left(2 \right)}} + 4
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = 3log2(x+5)-4
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