MA 137: Calculus I for the Life Sciences
David Murrugarra
Department of Mathematics,
University of Kentucky
http://www.ms.uky.edu/~ma137/
Spring 2017
David Murrugarra (University of Kentucky)
MA 137: Section 1.1
Spring 2017
1/8
Section 1.2: Logarithmic Functions
Definition
The inverse of f (x) = ax is called the logarithm with base a and is
written
f −1 (x) = loga x.
David Murrugarra (University of Kentucky)
MA 137: Section 1.1
Spring 2017
2/8
Section 1.2: Logarithmic Functions
Definition
The inverse of f (x) = ax is called the logarithm with base a and is
written
f −1 (x) = loga x.
That means,
y = loga x if and only if ay = x
David Murrugarra (University of Kentucky)
MA 137: Section 1.1
Spring 2017
2/8
Section 1.2: Logarithmic Functions
Definition
The inverse of f (x) = ax is called the logarithm with base a and is
written
f −1 (x) = loga x.
That means,
y = loga x if and only if ay = x
or
aloga x = x.
David Murrugarra (University of Kentucky)
MA 137: Section 1.1
Spring 2017
2/8
Section 1.2: Logarithmic Functions
Definition
The inverse of f (x) = ax is called the logarithm with base a and is
written
f −1 (x) = loga x.
That means,
y = loga x if and only if ay = x
or
aloga x = x.
Rule (Properties of Logarithms)
1
loga 1 = 0
2
loga a = 1
3
loga ax = x
David Murrugarra (University of Kentucky)
MA 137: Section 1.1
Spring 2017
2/8
Section 1.2: Natural Logarithms
Definition
The logarithm with base e is called the natural logarithm and is
written
ln x = loge x.
That means,
y = ln x if and only if ey = x
David Murrugarra (University of Kentucky)
MA 137: Section 1.1
Spring 2017
3/8
Section 1.2: Natural Logarithms
Definition
The logarithm with base e is called the natural logarithm and is
written
ln x = loge x.
That means,
y = ln x if and only if ey = x
or
eln x = exp(ln x) = x.
Rule (Properties of Natural Logarithms)
1
ln 1 = 0
2
ln e = 1
3
ln ex = x
David Murrugarra (University of Kentucky)
MA 137: Section 1.1
Spring 2017
3/8
Section 1.2: Common Logarithms
Definition
The logarithm with base 10 is called the common logarithm and is
written
log x = log10 x.
That means,
y = log x if and only if 10y = x
David Murrugarra (University of Kentucky)
MA 137: Section 1.1
Spring 2017
4/8
Section 1.2: Common Logarithms
Definition
The logarithm with base 10 is called the common logarithm and is
written
log x = log10 x.
That means,
y = log x if and only if 10y = x
or
10log x = x.
Rule (Properties of Natural Logarithms)
1
log 1 = 0
2
log 10 = 1
3
log 10x = x
David Murrugarra (University of Kentucky)
MA 137: Section 1.1
Spring 2017
4/8
Section 1.2: Laws of Logarithms
Rule (Properties of Logarithmic Functions)
Let a be a positive number with a 6= 1. Let A, B, and C any real
numbers with A > 0 and B > 0. Then
1
loga (AB) = loga A + loga B
David Murrugarra (University of Kentucky)
MA 137: Section 1.1
Spring 2017
5/8
Section 1.2: Laws of Logarithms
Rule (Properties of Logarithmic Functions)
Let a be a positive number with a 6= 1. Let A, B, and C any real
numbers with A > 0 and B > 0. Then
1
loga (AB) = loga A + loga B
2
loga ( BA ) = loga A − loga B
David Murrugarra (University of Kentucky)
MA 137: Section 1.1
Spring 2017
5/8
Section 1.2: Laws of Logarithms
Rule (Properties of Logarithmic Functions)
Let a be a positive number with a 6= 1. Let A, B, and C any real
numbers with A > 0 and B > 0. Then
1
loga (AB) = loga A + loga B
2
loga ( BA ) = loga A − loga B
3
loga (AC ) = C loga A.
David Murrugarra (University of Kentucky)
MA 137: Section 1.1
Spring 2017
5/8
Section 1.2: Change of Base
Rule
logb x =
loga x
loga b
Proof.
Let y = logb x, then by = x. Then apply loga (.) to both sides
David Murrugarra (University of Kentucky)
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Spring 2017
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Section 1.2: Change of Base
Rule
logb x =
loga x
loga b
Proof.
Let y = logb x, then by = x. Then apply loga (.) to both sides
loga (by ) = loga x ↔ y loga (b) = loga x
Then
David Murrugarra (University of Kentucky)
MA 137: Section 1.1
Spring 2017
6/8
Section 1.2: Change of Base
Rule
logb x =
loga x
loga b
Proof.
Let y = logb x, then by = x. Then apply loga (.) to both sides
loga (by ) = loga x ↔ y loga (b) = loga x
Then
y = logb x =
David Murrugarra (University of Kentucky)
loga x
loga (b)
MA 137: Section 1.1
Spring 2017
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Section 1.2: Change of Base
Rule (Change of Base from e)
loga x =
David Murrugarra (University of Kentucky)
ln x
ln a
MA 137: Section 1.1
Spring 2017
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Section 1.2: Change of Base
Rule (Change of Base from e)
loga x =
ln x
ln a
Rule (Change of Base from 10)
loga x =
David Murrugarra (University of Kentucky)
log x
log a
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Spring 2017
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Section 1.2: Trigonometric Functions
Definition (Periodic Functions)
A function f (x) is periodic if there is a positive constant a such that
f (x + a) = f (x)
for all x in the domain. If a is the smallest number with this property we
call it the period of f (x).
David Murrugarra (University of Kentucky)
MA 137: Section 1.1
Spring 2017
8/8
Section 1.2: Trigonometric Functions
Definition (Periodic Functions)
A function f (x) is periodic if there is a positive constant a such that
f (x + a) = f (x)
for all x in the domain. If a is the smallest number with this property we
call it the period of f (x).
Example
1
sin x has period 2π.
2
cos x has period 2π.
3
tan x has period π.
David Murrugarra (University of Kentucky)
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Spring 2017
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