ƒ(x) = **b**^{x}

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 (b^{0}=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.

2^{x} 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%.

3^{x} 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%.

1.5^{x} is the "increase by 50%" function.

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

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

.5^{x}is 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, 1^{x}=1 for all x, so it's the horizontal line y=1,
which isn't considered to be exponential.

Base * e* ≈2.718281828...

Examples of exponential functions

*e ^{x}* inverse is the natural logarithm function:

b

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.