Exponential Functions
Ÿ Geometric Growth - where each stage is a constant multiple of the one before.
Ÿ Example: A population of 100 bacteria doubles every hour
t=0
100
100
100
or
or
1
200
2*100
2*100
2
400
2*2*100
22 * 100
3
800
2*2*2*100
23 * 100
4 hrs
1600 etc.
2*2*2*2*100
24 * 100
By induction t hours ® 2t * 100 bacteria
Ÿ Example: $300 is invested and doubles every 7 years
t=0
300
7
2*300
14
21yrs
22 * 300 23 * 300 etc.
So, if t=1 then the investment is 217 * $300 and in general for t years we have 2t7 *$300
Ÿ This works for diminishing functions as well.
Example: My $20000 car loses 1/10 of its value every 3 months
t=0
20000
3 months
0.9*20000
1 year
0.94 * 20 000
2 years
0.98 * 20 000 etc.
My car then is worth J 10 N * $20000 after t years
9 4t
Ÿ Properties of exponentials
PowerExpand @Ha bLx D H* Ha bLx = ax bx *L
ax bx
PowerExpand @ax ay D H* ax ay = ax+y *L
ax+y
PowerExpand B
ax
ay
F H* ax  ay = ax-y *L
ax-y
PowerExpand AHax Ly E H* Hax Ly = ax
a
xy
y
*L
2
lecture4-exponentials&inverses.nb
Ÿ General Exponential Form
The basic exponential function is y = f(x) = bx . The steepness (or growth rate) is determined by changing b, as mentioned in the
previous lecture. Using the last property of exponentials above it is easy to show that this is the same as scaling horizontally by
some constant. The only other thing that typically needs to be controlled is the y-intercept, which can be taken care of with
vertical scaling. So the general form would be y = f(x) = a bx . This is the function most calculators try to fit when they do
exponential regression.
Alternatively, most scientists prefer not to mess around with the base and instead stick to one universal constant, referred to as
the "natural exponent" ã = 2.71828.... In this case the only way to change the growth rate and keep the base the same is to use
horizontal scaling (for exponential decay the scaling constant would be negative, effectively flipping the function horizontally).
This results in a general form of y = f(x) = a ãc x . This is the form that Excel uses when it does an exponential trendline. The
distinguishing characteristic of using ã as a base is that y = ãx has a tangent slope of exactly 1 at x = 0 (whereas generally y = bx
has a different slope at x = 0 - we will calculate exactly what later)
Inverse Functions
Ÿ In General - y = f(x) • x = f -1 HyL
Ÿ If there is some combination of calculations you use to produce a y from an x then hopefully you can reverse those calculations
(opposite calculation in reverse order) to get back the original x (sort of an "undo" operation). The process of calculating y from x
is known as the function y = f(x). The reversed calculation that produces x is known as the inverse function x = f -1 H yL
We can think of the inverse as a "cancellation" operation. This leaves us with the following identities
è f-1 Hf HxLL = x
è f If-1 HxLM = x
Ÿ For functions using the generic variables x and y we like to use x for the input variable and y for the output, so when we describe
the inverse function we switch the x and y above to get y = f -1 HxL.
Technically we can only do this if the original function was what we call "one-to-one". Meaning that not only is there only one y
value for every x, but there is only one x value for every y (otherwise the inverse is ambiguous and not really a function).
This also leaves us with the result that the domain of the new function (the inverse) is the same as the range of the original
function and vice-versa. Since it is mathematically easier to figure out the domain of a function, it becomes advantageous to use
the inverse of a function to figure out the range.
However, if we are discussing a function where the variables actually mean something (such as t for time and d for distance),
when we find the inverse function we generally don't switch the meaning of the variables since it would just confuse things.
Ÿ To actually find the inverse function, we can refer to the four different representations of a function.
Verbally, if I describe how far an object falls in a given amount of time I can turn the emphasis around and refer to the amount of
time it takes to fall a given distance.
Numerically, when we would normally use a table to look up a y value for a given x, we can just switch the labels or rows/columns around.
lecture4-exponentials&inverses.nb
è f HxL Þ
x 0 1 2 3 4
y 0 3 12 27 48
è f-1 HxL Þ
x 0 3 12 27 48
y 0 1 2 3 4
Visually, this amounts to switching the x and y axes on the graph. However, conventionally x is always the horizontal axis, so in
reality we end up flipping the function or curve across the diagonal y = x
:PlotA9x2 , x=, 8x, 0, 2<, PlotRange ® 880, 2<, 80, 2<<, AspectRatio ® AutomaticE,
PlotB:
x , x>, 8x, 0, 2<, PlotRange ® 880, 2<, 80, 2<<, AspectRatio ® AutomaticF>
2.0
1.5
:1.0
,
0.5
0.0
0.0
0.5
1.0
1.5
2.0
2.0
1.5
>
1.0
0.5
0.0
0.0
0.5
1.0
1.5
2.0
Algebraically what we do is solve the equation y = f(x) for x and then switch the two variables
3
4
lecture4-exponentials&inverses.nb
Algebraically what we do is solve the equation y = f(x) for x and then switch the two variables
eq1 = y Š 2 x - 3;
Reduce@eq1, xD
3
xŠ
y
+
2
2
1-2x
eq2 = y ==
;
3+x
Reduce@eq2, xD@@2DD
1-3y
xŠ
2+y
Ÿ Exponential Inverses Specifically - y = ax • x = loga y
Ÿ Basically this means that the logarithm base a of some number is whatever power you have to raise a to in order to get that number
This leads to the same identities for cancellation
loga Hax L = x
aloga x = x
Ÿ As a result we can also demonstrate a number of properties of logarithms that are related to the properties of exponentials already
discussed
loga Hx yL = loga x + loga y
loga Hxr L = r loga x
lecture4-exponentials&inverses.nb
Ÿ A particular logarithm that is used frequently is the "natural" logarithm, using ã as the base, logã x = ln x
8Plot@8ãx , x<, 8x, - 2, 5<, PlotRange ® 88- 2, 5<, 80, 5<<, AspectRatio ® AutomaticD,
Plot@8Log@xD, x<, 8x, 0, 5<, PlotRange ® 880, 5<, 8- 2, 5<<, AspectRatio ® AutomaticD<
5
4
3
:
,
2
1
-2
-1
0
1
2
3
4
5
5
4
3
2
>
1
0
-1
-2
1
2
3
4
5
5
6
lecture4-exponentials&inverses.nb
Ÿ The logarithm allows us to solve equations that involve exponentials by taking the logarithm of both sides of the equation (and vice
versa with equations that involve logarithms).
è Example of an exponential equation
10
f@x_D :=
1 + 9 ã-2 x
Plot@f@xD, 8x, 0, 10<D
10
9
8
7
6
2
4
6
8
eq1 = y Š f@xD;
eq1 = ExpandAMapAI1 + 9 ã-2 x M ð &, eq1EE
y + 9 ã-2 x y Š 10
eq1 = Map@ð - y &, eq1D
9 ã-2 x y Š 10 - y
ð
eq1 = MapB
&, eq1F
9y
ã-2 x Š
10 - y
9y
eq1 = PowerExpand @ Map @Log@ðD &, eq1DD
- 2 x Š - Log@9D + Log@10 - yD - Log@yD
Reduce@Simplify@eq1D, xD
10 - y
1
xŠ-
LogB
2
9
F+
Log@yD
2
10
lecture4-exponentials&inverses.nb
10 - x
1
PlotB-
LogB
2
9x
7
F, 8x, 0, 10<F
4
3
2
1
2
4
6
8
10
-1
è Example of a logarithmic equation
eq2 = Log@2D Š 3 Log@xD - 1;
eq2 = MapAãð &, eq2E
x3
2Š
ã
Reduce@eq2D@@2DD
x Š H2 ãL13
Ÿ To evaluate logarithms, you can either solve it exactly using the properties of logarithms and exponentials to simplify a logarithmic
expression, or approximate it using your calculator and the "change of base formula"
è Example of simplification (when the base and the operand have another base in common)
x = log4 8 ® 4x = 8 ® I22 M = 23 ® 22 x = 23 ® 2 x = 3 ® x = 3  2
x
è Example of "change of base" - loga x =
x = log2 10
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x=
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3.32193
23.32193
10.
ln x
ln a