Exponential functions.

ƒ(x) = bx

b is the base, any/every positive real number: b>0. There are an infinite number of different exponential functions, one for every positive real number. The variable x is the exponent. These are not polynomial functions.

Each exponential function's graph in the XY Cartesian plane is an exponential curve.
It crosses the y axis at 1 (b0=1 for all b); i.e. the y-intercept is 1.
It does not cross the X axis (y is never 0 or negative); i.e. X axis is a horizontal asymptote.

If b>1, the curve is increasing (goes up to the right):

Usually in applications, the negative x values don't have meaning and so aren't used.

2x is the doubling function. For every one more unit of X, the y doubles (i.e. if x increases by 1, y doubles (f(x+1)=f(x)*2)). It increases by 100%.

3x is the tripling function. For every one more unit of X, the y triples (i.e. if x increases by 1, y triples (f(x+1)=f(x)*3)). It increases by 200%.

1.5x is the "increase by 50%" function.

The 1.1x above might look flat, but eventually every exponential function of base b>1 goes "exponential", this is the essence of exponential growth. 1.1x represents 10% compounded growth.

If b<1, the curve is decreasing (goes down to the right):

.5xis the halving function. For every one more unit of X, the y halves (i.e. if x increases by 1, y halves (f(x+1)=f(x)/2)). It decreases by 50%.

If b=1, 1x=1 for all x, so it's the horizontal line y=1, which isn't considered to be exponential.

Base e ≈2.718281828...
ex is the exponential function.

More about e

Examples of exponential functions

ex inverse is the natural logarithm function: ln

bx=eln(b)x     2x=eln(2)x=e.6931x     3x=eln(3)x=e1.098x

Derivative of ln: (ln x)' = 1/x
Integral of ln: ∫ln(x) = x ln(x)- x = x(ln x -1)

"Naturalness": slope of tangent line to ln(x) at (1,0) is 1.