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log(4*x^3+e^x)

  • How to use it?

  • Graphing y =:
  • (x^2+5)/(x-3)
  • 2x^3-3x^2+5
  • x^4-2x^2-8
  • -x^4+2x^2+3

  • Identical expressions

  • log(four *x^ three +e^x)
  • logarithm of (4 multiply by x cubed plus e to the power of x)
  • logarithm of (four multiply by x to the power of three plus e to the power of x)
  • log(4*x3+ex)
  • log4*x3+ex
  • log(4*x³+e^x)
  • log(4*x to the power of 3+e to the power of x)
  • log(4x^3+e^x)
  • log(4x3+ex)
  • log4x3+ex
  • log4x^3+e^x

  • Similar expressions

  • log(4*x^3-e^x)
Plotting a function graph/ log(4*x^3+e^x)

Graphing y = log(4*x^3+e^x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          /   3    x\
f(x) = log\4*x  + e /
f(x)=log⁡(4x3+ex)f{\left(x \right)} = \log{\left(4 x^{3} + e^{x} \right)} 
f = log(4*x^3 + E^x)
The graph of the function
02468-8-6-4-2-101020-10
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:
log⁡(4x3+ex)=0\log{\left(4 x^{3} + e^{x} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Numerical solution
x1=0x_{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(4*x^3 + E^x).
log⁡(4⋅03+e0)\log{\left(4 \cdot 0^{3} + e^{0} \right)}
The result:
f(0)=0f{\left(0 \right)} = 0
The point:
(0, 0)
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
12x2+ex4x3+ex=0\frac{12 x^{2} + e^{x}}{4 x^{3} + e^{x}} = 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
24x−(12x2+ex)24x3+ex+ex4x3+ex=0\frac{24 x - \frac{\left(12 x^{2} + e^{x}\right)^{2}}{4 x^{3} + e^{x}} + e^{x}}{4 x^{3} + e^{x}} = 0
Solve this equation
The roots of this equation
x1=77.198731330601x_{1} = 77.198731330601
x2=77.0124850969537x_{2} = 77.0124850969537
x3=60.6525896883197x_{3} = 60.6525896883197
x4=77.0838695174728x_{4} = 77.0838695174728
x5=100x_{5} = 100
x6=64.5261565642568x_{6} = 64.5261565642568
x7=6.04601858923138x_{7} = 6.04601858923138
x8=78x_{8} = 78
x9=70.3757428307212x_{9} = 70.3757428307212
x10=94.25x_{10} = 94.25
x11=74.3855050604014x_{11} = 74.3855050604014
x12=72.3229314979781x_{12} = 72.3229314979781
x13=51.1071218018637x_{13} = 51.1071218018637
x14=80x_{14} = 80
x15=98x_{15} = 98
x16=52.9946996224697x_{16} = 52.9946996224697
x17=62.5865072888081x_{17} = 62.5865072888081
x18=77.1862034519915x_{18} = 77.1862034519915
x19=96x_{19} = 96
x20=77.8203482568241x_{20} = 77.8203482568241
x21=77.2141744553657x_{21} = 77.2141744553657
x22=56.8055953511733x_{22} = 56.8055953511733
x23=77.3618108426466x_{23} = 77.3618108426466
x24=45.5503908345065x_{24} = 45.5503908345065
x25=54.8948687600961x_{25} = 54.8948687600961
x26=84x_{26} = 84
x27=77.0258369002429x_{27} = 77.0258369002429
x28=82x_{28} = 82
x29=40.2805348535545x_{29} = 40.2805348535545
x30=90x_{30} = 90
x31=47.3809756966331x_{31} = 47.3809756966331
x32=43.7493508424981x_{32} = 43.7493508424981
x33=66.4707529956174x_{33} = 66.4707529956174
x34=77.0566778668501x_{34} = 77.0566778668501
x35=68.419631664462x_{35} = 68.419631664462
x36=77.1427592390923x_{36} = 77.1427592390923
x37=77.1877987770559x_{37} = 77.1877987770559
x38=86x_{38} = 86
x39=88x_{39} = 88
x40=58.7252674784738x_{40} = 58.7252674784738
x41=77.6251821417523x_{41} = 77.6251821417523
x42=92x_{42} = 92
x43=0x_{43} = 0
x44=41.9873266562552x_{44} = 41.9873266562552
x45=49.234743592684x_{45} = 49.234743592684

Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
[90,∞)\left[90, \infty\right)
Convex at the intervals
(−∞,0]\left(-\infty, 0\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞log⁡(4x3+ex)=∞\lim_{x \to -\infty} \log{\left(4 x^{3} + e^{x} \right)} = \infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞log⁡(4x3+ex)=∞\lim_{x \to \infty} \log{\left(4 x^{3} + e^{x} \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(4*x^3 + E^x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(log⁡(4x3+ex)x)=0\lim_{x \to -\infty}\left(\frac{\log{\left(4 x^{3} + e^{x} \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(log⁡(4x3+ex)x)=1\lim_{x \to \infty}\left(\frac{\log{\left(4 x^{3} + e^{x} \right)}}{x}\right) = 1
Let's take the limit
so,
inclined asymptote equation on the right:
y=xy = x
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:
log⁡(4x3+ex)=log⁡(−4x3+e−x)\log{\left(4 x^{3} + e^{x} \right)} = \log{\left(- 4 x^{3} + e^{- x} \right)}
- No
log⁡(4x3+ex)=−log⁡(−4x3+e−x)\log{\left(4 x^{3} + e^{x} \right)} = - \log{\left(- 4 x^{3} + e^{- x} \right)}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = log(4*x^3+e^x)
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