Exponential functions

2⁵, 5²
x², x⁵
2^x, 5^x

Exponential functions.

ƒ(x) = b^x

b is the base, any/every positive real number: b>0. (0 as a base would have 0⁰ and e.g. 0^-1=1/0 problems) (negative bases would have e.g. -1^(1/2)=√-1 complex numbers).
There are an infinite number of different exponential functions, one for each positive real number.
The variable x is the exponent.
Domain is ℝ.
NB. These are not polynomial functions.

Each exponential function's graph in the XY Cartesian plane is an exponential curve, continuous and smooth.
It crosses the y axis at 1 (f(0)=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 one-sided horizontal asymptote (the other end goes to +∞).
The range is (0,∞), i.e. all positive real numbers.
No X-intercepts, no turning points, no extrema, no "wiggle", no domain issues.

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

This is exponential growth.
The larger the base b, the steeper/faster the increase.
Usually in applications, the negative x values don't have meaning and so aren't used.
b is the growth factor.
f(-1)=1/b f(0)=1 f(1)=b

2^x is the doubling function. For every one more unit of X, the y doubles (i.e. x increases by 1, y doubles (f(x+1)=f(x)*2); i.e. x→x+1, y→2y). It increases by 100%. (NB. the linear function f(x)=2x is "increase by 2"; i.e. x→x+1, y→y+2)
Powers of 2 worksheet

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

1.5^x is the "increase by 50%" function. For every one more unit of X, the y increases by half its current value (i.e. x increases by 1, y (f(x+1)=f(x)*1.5); i.e. x→x+1, y→y+½y)).

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. (It's actually "exponential" everywhere; it just doesn't look like it is at small x values.) 1.1^x represents 10% (compounded) growth. If the horizontal axis is time, you have to give it some time before it "rockets" up. The world has been on such a curve since the start of the Industrial Revolution: science/technology, economy, population, wealth.

Exponential: the bigger it is, the more it grows. It increases faster/more as x increases.
Every increasing exponential function eventually surpasses every increasing polynomial function.

Base e ≈2.718281828 AKA Euler's number
e^x is the exponential function. Sometimes denoted: exp(x)
The natural exponential function.

e^-1=1/e=0.3678

e is the base for continuous growth/decay applications.
Perhaps the most important single function.

More about e

Convert any base b expression to base e: b^x = e^{x ln b}

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

If b<1, (i.e. between 0 and 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. x increases by 1, y halves (f(x+1)=f(x)/2); i.e. x→x+1, y→y/2). It decreases by 50%.
It's the reflection of f(x)=2^x across the y-axis.

Exponential decay curves:

Negative exponent:    b^-x    = 1/b^x = (1/b)^x
Exs. 2^-x = (1/2)^x    decaying
e^-x = (1/e)^x    decaying!

b^x and b^-x are reflections of each other across the Y-axis.
b^x and (1/b)^x (i.e. reciprocals) are reflections of each other across the Y-axis.

Flip over either or both axes.
Shift left or right.
Shift up or down.
Stretch or compress (squash).
↓
General exponential functions: ƒ(x)= a·b^kx-d + c
a: If >1, stretchs the graph up; if <1, compresses down. Will be y-intercept / initial value (unless c). If negative, flips graph over X-axis.
k: If >1, steepens the curve; if <1, flattens/broadens it. If negative, switches between increasing and decreasing exponential (see below), flipping the graph over the Y-axis.
d: Horizontal shift left/right.
c: Vertical shift raises or lowers the resulting graph.
Always has a one-sided horizontal asymptote. And restricted range.

Inverse is a log function:
ƒ^-1(x)= (log_b((x-c)/a) + d) / k

Derivative: (ab^kx-d + c)' = akb^kx ln b
(ae^kx-d + c)' = ake^kx
Integral: ∫ab^kx-d + c = ab^kx / (k ln b) + cx
∫ae^kx-d + c = ae^kx / k + cx

Exponential with positive coefficient k:	Exponential with negative coefficient k:

The larger the positive coefficient k, the faster the growth.	The \|larger\| the negative coefficient k, the faster the decay.

In applications, the most common variable is time. #2 is length?
ƒ(t) = ae^kt, where a is the starting/initial amount and k is the growth (k>0) or decay (k<0) rate.
(0,a) is y-intercept.

Doubling time of increasing exponential functions:
ƒ(t)= a·b^kt has a doubling time t_d = (ln 2) / (k ln b)
ƒ(t)= a·e^kt has a doubling time t_d = (ln 2) / k ≈ .6931/k
The function's value doubles in (every/any) t_d time units.
If at time t_now the function value is y, then at time t_now+t_d the function value is 2y.
The time between every doubling (or halving) is the same.
Exponential functions are the only kind of function that have this property.
Halving time of decreasing exponential functions is (ln (1/2)) / k = -(ln 2) / k ≈ -.6931/k

Exponential functions worksheet
Exponential exercises answers
Exponential/Log equations

Compounded Interest worksheet
Radioactive decay
Newton's Law of cooling/warming calculations
Logistic growth Logistic exercises

e^x inverse is the natural logarithm function: ln

ln(exp(x))=x and exp(ln(x))=x
b^x=e^{x ln b} 2^x=e^{x ln 2}=e^.6931x 3^x=e^ln(3)x=e^1.098x
e^ax=(e^a)^x

Two points (x₁,y₁) and (x₂,y₂) [both y's positive or both y's negative] determine an exponential function y=ab^x.
a=y₁ / (y₂/y₁)^{x₁/(x₂-x₁)}
b=(y₂/y₁)^{1/(x₂-x₁)}
Positive y's: a>0
Negative y's: a<0