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 every 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 real numbers >0.
No X-intercepts, no turning points, no extrema, no "wiggle", no domain issues.
f(-1)=1/b f(0)=1 f(1)=b
If b>1, the curve is increasing (goes up to the right):
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.
2x 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)).
It increases by 100%. (NB. the linear function f(x)=2x is "increase by 2")
Powers of 2 worksheet
3x 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)). 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.
b is the growth factor.
Exponential: the bigger it is, the more it grows. It increases faster as x increases.
Every increasing exponential function eventually surpasses every increasing polynomial function.
Base e ≈2.718281828
ex is the exponential function. Sometimes denoted: exp(x)
e-1=1/e=0.3678
bx = ex ln b
If b=1, 1x=1 for all x, so it's the horizontal line y=1, which isn't considered to be exponential.
If b<1, the curve is decreasing (goes down to the right):
.5x 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)).
It decreases by 50%.
It's the reflection of f(x)=2x across the y-axis.
Negative exponent:
b-x = 1/bx = (1/b)x
Exs. 2-x = (1/2)x decaying
e-x = (1/e)x decaying!
bx and b-x are reflections of each other across the Y-axis.
Negative exponent with coefficient:
The larger the coefficient, the faster the decay.
General exponential functions:
ƒ(x)= a·bkx-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.
c raises or lowers the resulting graph.
Has a one-sided horizontal asymptote.
Derivative: akbkx ln b
Integral: abkx / (k ln b) + cx
Doubling time of increasing exponential functions:
ƒ(t)= a·bkt
has a doubling time td = (ln 2) / (k ln b)
ƒ(t)= a·ekt
has a doubling time td = (ln 2) / k ≈ .6931/k
The function's value doubles in (every/any) td time units.
If at time tnow the function value is y,
then at time tnow+td the function value is 2y.
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
Compounded Interest
worksheet
Radioactive decay
Newton's Law of cooling/warming calculations
Logistic growth
Logistic exercises
ex inverse is the natural logarithm function: ln
ln(exp(x))=x and exp(ln(x))=x
bx=ex ln b
2x=ex ln 2=e.6931x
3x=eln(3)x=e1.098x
eax=(ea)x
Two points (x1,y1) and (x2,y2)
determine an exponential function y=abx.
a=y1 / (y2/y1)x1/(x2-x1)
b=(y2/y1)1/(x2-x1)