Mathematical tables: exponential function formulas
Table show common formulas related to exponential function.

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Various forms of function formulas#

NameFormulaLegend
Exponential function in general formShow sourcey=axy=a^{x}
  • yy - function value (the function value at single point x, often marked as f(x)),
  • xx - function argument (called also independent value),
  • aa - base of the exponential function.
Exponential function with base e (often written as exp(x))Show sourceexp(x)=exexp(x)=e^{x}
  • exp(x)exp(x) - value of exponent function,
  • xx - function argument (called also independent value),
  • ee - number e (mathematical constant, base of natural logarithm).
Homographic function in general formShow sourcey=ax+bcx+dy=\frac{a \cdot x+b}{c \cdot x+d}
  • yy - function value (the function value at single point x, often marked as f(x)),
  • xx - function argument (called also independent value),
  • a, b, c, d - homographic function coefficients (parameters defining concrete homographic function, c ≠ 0).
Function b/xShow sourcey=bxy=\frac{b}{x}
  • yy - value of b/x function (the value of f(x)=b/x function for given x, parameters a,d are zero, parameter c is 1),
  • xx - function argument (called also independent value),
  • b - coefficient b.
Linear function in slope-intercept formShow sourcey=ax+by=a \cdot x+b
  • yy - function value (the function value at single point x, often marked as f(x)),
  • xx - function argument (called also independent value),
  • aa, bb - linear function coefficients (slope and free parameter).
Linear function in point-slope formShow sourcey=a(xx0)+y0y=a\left(x-x_0\right)+y_0
  • yy - function value (the function value at single point x, often marked as f(x)),
  • xx - function argument (called also independent value),
  • aa - slope (number that describes the direction and the steepness of the line, sometimes is called gradient),
  • x0x_0, y0y_0 - point coordinates.
Linear function in constant-slope formShow sourceyy0xx0=y1y0x1x0\frac{y - y_0}{x - x_0} = \frac{y_1 - y_0}{x_1 - x_0}
  • yy - function value (the function value at single point x, often marked as f(x)),
  • xx - function argument (called also independent value),
  • x0x_0, y0y_0 - coordinates of the first point,
  • x1x_1, y1y_1 - coordinates of the second point.
Zero of the linear function from constant-slope formShow sourcex=y0(x1x0)y1y0+x0x=\frac{y_0 \cdot \left(x_1-x_0\right)}{y_1-y_0}+x_0
  • xx - zero of the function (argument for which the function has a value of zero, its a solution of f(x) = 0 equation),
  • x0x_0, y0y_0 - coordinates of the first point,
  • x1x_1, y1y_1 - coordinates of the second point.
Quadratic function in standard formShow sourcey=ax2+bx+cy=a \cdot x^{2}+b \cdot x+c
  • yy - function value (the function value at single point x, often marked as f(x)),
  • xx - function argument (called also independent value),
  • aa, bb, cc - quadratic function coefficients (numbers just before x2, x and free parameter).
Quadratic function in factored formShow sourcey=a(xx1)(xx2)y=a\left(x-x_1\right) \cdot \left(x-x_2\right)
  • yy - function value (the function value at single point x, often marked as f(x)),
  • xx - function argument (called also independent value),
  • aa - coefficient before power of two (number just before x2),
  • x1x_1, x2x_2 - function zero points (arguments, for which function has value of zero, solutions of the f(x)=0 equation).
Quadratic function in vertex formShow sourcey=a(xh)2+ky=a\left(x-h\right)^{2}+k
  • yy - function value (the function value at single point x, often marked as f(x)),
  • xx - function argument (called also independent value),
  • aa - coefficient before power of two (number just before x2),
  • hh, kk - coordinates of the parabola vertex (at this point function reaches its local extremum).

Function properties#

NameFormulaLegend
Product of two exponential functionsShow sourceaxay=ax+ya ^ {x} \cdot a ^ {y} = a ^ {x + y}
  • xx - the first argument,
  • yy - the second argument,
  • aa - base of the exponential function.
Quotient of two exponential functionsShow sourceaxay=axy\frac{a ^ {x}}{a ^ {y}} = a ^ {x - y}
  • xx - the first argument,
  • yy - the second argument,
  • aa - base of the exponential function.

Derivatives and integrals#

NameFormulaLegend
Derivative of general exponential functionShow sourceddxax=axln(a)\frac{d}{d x} a^{x} = a^{x} \ln(a)
  • xx - function argument (called also independent value),
  • aa - base of the exponential function.
Derivative of exponential function with e base, derivative of exp(x) functionShow sourceddxex=exln(e)=ex\frac{d}{d x} e^{x} = e^{x} \ln(e) = e^{x}
Indefinite integral of general exponential functionShow sourceaxdx=axln(a)\int a^{x} dx = \frac{a^{x}}{\ln(a)}
  • xx - function argument (called also independent value),
  • aa - base of the exponential function.
Indefinite integral of exponential function with e base, integral of exp(x) functionShow sourceexdx=ex\int e^{x} dx = e^{x}

Some facts#

  • Exponential function is a function that can be presented in the form:
    y=axy=a^{x}
    where:
    • yy - function value (the function value at single point x, often marked as f(x)),
    • xx - function argument (called also independent value),
    • aa - base of the exponential function.
  • A special case is the exponential function of the base e (→ see the number e):
    exp(x)=exexp(x)=e^{x}
  • The exponential function with the base e is often denoted as exp (x), which we read as exponent of x.
  • The inverse function for the exponential one is logarithmic function. In particular for the function exp(x) (the base is number e) the inverse function is natural logarithm.
  • The exponential function has no zero poins. Its all values ​​are located above the OX axis (all function values are positive).
  • Depending on the base a, we can distinguish three scenarios:
    • base is less than one (a <1) - function is decreasing,
    • base is greater than one (a > 1) - the function is inreasing,
    • base is equal to zero (a = 0) - the function is reduced to constant function.

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