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

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# Various forms of function formulas#

 Name Formula Legend Exponential function in general form Show source$y=a^{x}$ $y$ - function value (the function value at single point x, often marked as f(x)),$x$ - function argument (called also independent value),$a$ - base of the exponential function. Exponential function with base e (often written as exp(x)) Show source$exp(x)=e^{x}$ $exp(x)$ - value of exponent function,$x$ - function argument (called also independent value),$e$ - number e (mathematical constant, base of natural logarithm). Homographic function in general form Show source$y=\frac{a \cdot x+b}{c \cdot x+d}$ $y$ - function value (the function value at single point x, often marked as f(x)),$x$ - function argument (called also independent value),a, b, c, d - homographic function coefficients (parameters defining concrete homographic function, c ≠ 0). Function b/x Show source$y=\frac{b}{x}$ $y$ - 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),$x$ - function argument (called also independent value),b - coefficient b. Linear function in slope-intercept form Show source$y=a \cdot x+b$ $y$ - function value (the function value at single point x, often marked as f(x)),$x$ - function argument (called also independent value),$a$, $b$ - linear function coefficients (slope and free parameter). Linear function in point-slope form Show source$y=a\left(x-x_0\right)+y_0$ $y$ - function value (the function value at single point x, often marked as f(x)),$x$ - function argument (called also independent value),$a$ - slope (number that describes the direction and the steepness of the line, sometimes is called gradient),$x_0$, $y_0$ - point coordinates. Linear function in constant-slope form Show source$\frac{y - y_0}{x - x_0} = \frac{y_1 - y_0}{x_1 - x_0}$ $y$ - function value (the function value at single point x, often marked as f(x)),$x$ - function argument (called also independent value),$x_0$, $y_0$ - coordinates of the first point,$x_1$, $y_1$ - coordinates of the second point. Zero of the linear function from constant-slope form Show source$x=\frac{y_0 \cdot \left(x_1-x_0\right)}{y_1-y_0}+x_0$ $x$ - zero of the function (argument for which the function has a value of zero, its a solution of f(x) = 0 equation),$x_0$, $y_0$ - coordinates of the first point,$x_1$, $y_1$ - coordinates of the second point. Quadratic function in standard form Show source$y=a \cdot x^{2}+b \cdot x+c$ $y$ - function value (the function value at single point x, often marked as f(x)),$x$ - function argument (called also independent value),$a$, $b$, $c$ - quadratic function coefficients (numbers just before x2, x and free parameter). Quadratic function in factored form Show source$y=a\left(x-x_1\right) \cdot \left(x-x_2\right)$ $y$ - function value (the function value at single point x, often marked as f(x)),$x$ - function argument (called also independent value),$a$ - coefficient before power of two (number just before x2),$x_1$, $x_2$ - function zero points (arguments, for which function has value of zero, solutions of the f(x)=0 equation). Quadratic function in vertex form Show source$y=a\left(x-h\right)^{2}+k$ $y$ - function value (the function value at single point x, often marked as f(x)),$x$ - function argument (called also independent value),$a$ - coefficient before power of two (number just before x2),$h$, $k$ - coordinates of the parabola vertex (at this point function reaches its local extremum).

# Function properties#

 Name Formula Legend Product of two exponential functions Show source$a ^ {x} \cdot a ^ {y} = a ^ {x + y}$ $x$ - the first argument,$y$ - the second argument,$a$ - base of the exponential function. Quotient of two exponential functions Show source$\frac{a ^ {x}}{a ^ {y}} = a ^ {x - y}$ $x$ - the first argument,$y$ - the second argument,$a$ - base of the exponential function.

# Derivatives and integrals#

 Name Formula Legend Derivative of general exponential function Show source$\frac{d}{d x} a^{x} = a^{x} \ln(a)$ $x$ - function argument (called also independent value),$a$ - base of the exponential function. Derivative of exponential function with e base, derivative of exp(x) function Show source$\frac{d}{d x} e^{x} = e^{x} \ln(e) = e^{x}$ $x$ - function argument (called also independent value),$e$ - number e (mathematical constant, base of natural logarithm). Indefinite integral of general exponential function Show source$\int a^{x} dx = \frac{a^{x}}{\ln(a)}$ $x$ - function argument (called also independent value),$a$ - base of the exponential function. Indefinite integral of exponential function with e base, integral of exp(x) function Show source$\int e^{x} dx = e^{x}$ $x$ - function argument (called also independent value),$e$ - number e (mathematical constant, base of natural logarithm).

# Some facts#

• Exponential function is a function that can be presented in the form:
$y=a^{x}$
where:
• $y$ - function value (the function value at single point x, often marked as f(x)),
• $x$ - function argument (called also independent value),
• $a$ - base of the exponential function.
• A special case is the exponential function of the base e (→ see the number e):
$exp(x)=e^{x}$
• 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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