Download Exponential and Logarithmic Functions

Exponential and Logarithmic
Functions
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The line is the most important function in
mathematics
The exponential function is a very close
second.
Everything else is a distant third.
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Quadratics, polynomials, power functions, absolute
value, rational functions, etc.
Why is the exponential function so important?
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The exponential function measures “steady growth”
The exponential function measures reproduction.
● Any time that things are making copies of
themselves, and exponential function describes what
happens.
● Populations → Animals making more animals.
● Investment → Money making more money
● Kinematics → Motion making more motion
● Radiology → Radioactive decay is negative
reproduction (negative growth).
● Etc.
Why is the exponential function so
hard to understand?
It grows too fast.
●People oversimplify.
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People oversimplify
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The exponential function is NOT about
repeated multiplication
The exponential function is CLOSELY
RELATED to repeated multiplication.
Imagine a population of bacteria
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Start with one bacterium
Every five minutes, each bacterium splits,
creating two bacteria.
What does a graph of this population look like?
O
mins
5
mins
10
mins
15
mins
Bacteria Graph
“repeated multiplication” or “geometric growth”
Bacteria Graph
“repeated multiplication” or “geometric growth”
exponential growth
Exponential Growth
Occurs NOT JUST when every individual is reproducing,
But when EVERY FRACTION of every individual is reproducing
“repeated multiplication” or “geometric growth”
exponential growth
Exponential Growth
Exponential growth is used to APPROXIMATE situations like bacteria
“repeated multiplication” or “geometric growth”
exponential growth
Every Fraction
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Populations → Animals making more animals.
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Investment → Money making more money
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Kinematics → Motion making more motion
Reasoning about every fraction is very
complicated.
Repeated multiplication a lie, but it's a very
useful lie, one that's not far from the truth.
f(x)=2x
f(x)=2x
+1
*2
f(x)=2x
*2
+1
*2
+1
f(x)=2x
+1
*2
Going the other way
*1/2
-1
+1, *2
-1, *(1/2)
-1, *(1/2)
y=1, 1/2, 1/4, 1/8, 1/16, 1/32,...
-1, *(1/2)
Because I'm shrinking by halves,
I will get very close to y=0,
but I will never reach it.
-1, *(1/2)
Because I'm shrinking by halves,
I will get very close to y=0,
but I will never reach it.
This is called an “asymptote” at y=0
Asymptote
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A line that your function gets infinitely close to,
but never reaches.
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Horizontal
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Vertical
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Oblique (slanted)
Every function of the form f(x)=abx has a
horizontal asymptote at y=0.
Consider the function below:
Which of the following statements matches with this
function?
A) As x approaches infinity, f(x) approaches 0.
B) As x approaches negative infinity, f(x) approaches
0.
C) As x approaches infinity, f(x) approaches -4.
D) As x approaches negative infinity, f(x) approaches
-4.
E) None of the above
Solution
D) As x approaches negative infinity, f(x) approaches -4.
Alternate Solution
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3x is sort of like “+1 to x, *3 to y” as I'm going
right.
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3x is sort of like “-1 to x, *1/3 to y” as I'm going
left.
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As x gets bigger, 3x gets bigger and bigger.
As x gets smaller, 3x gets closer and closer to 0.
As x gets smaller, 3x-4 gets closer and closer to
-4.
D) As x approaches negative infinity, f(x) approaches -4.
Negative b
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If 3^x is like repeatedly multiplying by 3,
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What is 3^(-x) like?
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3^(-1)^(x)=(1/3)^x.
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3^(-x) is like repeatedly multiplying by (1/3)
Exponential Decay
Logarithms
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A logarithm is an inverse of an exponential
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(see inverse functions)
If I have the function f(x)=6x, then log67 means
“what number does x have to be so that 7=6x?”
Logarithm
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Logab
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Pronounced “Log base a of b”
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Means “the number (power) I have to raise a to
to get b.”
x=Logab means ax=b
Special logarithms
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Log(b) means Log10b
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ln(b) means Logeb, where e≈2.7182818284
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You only ever need one log function
Loga(b)=Log(b)/Log(a)=ln(b)/ln(a)
Why e?
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You'll find out more about e in studio.
When every individual is doubling every time unit, the
population fits f(x)=2x,
When the same process is updated not just every time
unit, but every fraction of every moment (and for every
fraction of every individual), the population fits f(x)=ex.
Because time is continuous, ex happens a lot more in
real life problems than 2x.
e is the most important number in mathematics (except
for 0 and 1). Much more important than π.
Domain of a Logarithmic function
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Exponential functions and logarithms are inverses.
Because you can never get a negative number OUT OF an
exponential, you can never put a negative number INTO a
logarithm.
Find the domain:
a)
b)
c)
d)
e)
A) x > 5/4
B) x < 5/4
C) x > -5/4
D) x < -5/4
E) None of the above
Solution
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The number you put into a logarithm must be
positive.
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5-4x must be positive.
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5-4x>0
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5>4x
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5/4>x
B) x < 5/4