5-1: Natural Logs, 1

5-1: Natural Logs, 1
Objectives:
1. To define the natural
logarithm and take its
derivative
Assignment:
• P. 329-331: 7-10, 19-33
odd, 37-41 odd, 45, 46,
49-51, 53, 54, 57, 59,
63, 65, 71, 72, 83, 93,
94, 96-98, 103, 104
Warm Up
Let’s say you had $1 to put into an account
for a year, and the interest rate was 100%.
How much money would you have in the
account if the interest was compounded
once (n = 1)? Twice (n = 2)? Will this
amount continue to grow as n increases?
What’s the best possible value of n? How
much money would you have in the bank at
the end of the year with this value of n?
Compound Interest
Consider an initial principal 𝑃 deposited in an account
that pays interest at an annual rate 𝑟 (expressed as
a decimal), compounded 𝑛 times per year. The
amount 𝐴 in the account after 𝑡 years is given by:
𝑟
𝐴=𝑃 1+
𝑛
Compounding refers to
adding the interest back to
the principal. This can be
𝑛𝑡
done yearly, monthly,
quarterly, daily, or even
continuously.
Warm Up
1
𝐴=1 1+
𝑛
𝑛
𝑛=1
𝑛=2
𝑛=4
𝑛 = 12
𝑛 = 365
𝑛 = 8,760
$2.00
$2.25
$2.44
$2.61
$2.71
$2.72
+.25
+.19
+.17
+.10
+.01
Limit!
Warm Up
1
𝐴=1 1+
𝑛
𝑛
Natural Base e
The natural base 𝑒 is an irrational number
such that
𝑛
1
lim 1 +
= 𝑒 ≈ 2.718281828459045 …
𝑛→∞
𝑛
•
•
Given a letter by
Leonhard Euler
(1707-1783)
Sometimes called
the Euler number
Natural Base e
The natural base 𝑒 is an irrational number
such that
𝑛
1
lim 1 +
= 𝑒 ≈ 2.718281828459045 …
𝑛→∞
𝑛
Like 𝜋, 𝑒 is a
transcendental
number since it is not
the root of any
polynomial equation
(Euler’s Formula, not Euler’s Arm)
Objective 1
You will be able to define
the natural logarithm and
take its derivative
Definition of Logarithm
Logarithm with Base 𝒃
For 𝑦 > 0, 𝑏 > 0, and 𝑏 ≠ 1, the logarithm with
base 𝑏 of 𝑦 is denoted as log 𝑏 𝑦 such that
log 𝑏 𝑦 = 𝑥
if and only if 𝑏 𝑥 = 𝑦
The base of the logarithm is the base of the exponential equation
Definition of Logarithm
Logarithm with Base 𝒃
For 𝑦 > 0, 𝑏 > 0, and 𝑏 ≠ 1, the logarithm with
base 𝑏 of 𝑦 is denoted as log 𝑏 𝑦 such that
log 𝑏 𝑦 = 𝑥
if and only if 𝑏 𝑥 = 𝑦
The answer to the logarithm is the exponent of the exponential
Definition of Logarithm
Logarithm with Base 𝒃
For 𝑦 > 0, 𝑏 > 0, and 𝑏 ≠ 1, the logarithm with
base 𝑏 of 𝑦 is denoted as log 𝑏 𝑦 such that
log 𝑏 𝑦 = 𝑥
if and only if 𝑏 𝑥 = 𝑦
The general logarithm function is an exponentproducing function. The log base b is the
exponent you have to put on b to get y.
Definition of Logarithm
Logarithm with Base 𝒃
For 𝑦 > 0, 𝑏 > 0, and 𝑏 ≠ 1, the logarithm with
base 𝑏 of 𝑦 is denoted as log 𝑏 𝑦 such that
log 𝑏 𝑦 = 𝑥
if and only if 𝑏 𝑥 = 𝑦
Logarithmic
Form
Exponential
Form
Exercise 1
Evaluate each logarithm.
1. log2 32
2. log10 1
3. log9 9
4. log1/5 25
5. log7 73
A Few Good Properties
Let 𝑏 be a positive real number with 𝑏 ≠ 1.
Since
𝑏 0 = 1,
log 𝑏 1 = 0
Since
𝑏𝑎 = 𝑏𝑎 ,
log 𝑏 𝑏 𝑎 = 𝑎
Since
𝑏1 = 𝑏,
log 𝑏 𝑏 = 1
Product Property
Product Property of
Logarithms
Let 𝑏, 𝑚, and 𝑛 be
positive real numbers
with 𝑏 ≠ 1.
log 𝑏 𝑚 ∙ 𝑛 =
log 𝑏 𝑚 + log 𝑏 𝑛
Product Property of
Exponents
𝑏 𝑚 ∙ 𝑏 𝑛 = 𝑏 𝑚+𝑛
“The log of a product
equals the sum of the
logs of the factors.”
Quotient Property
Quotient Property of
Logarithms
Let 𝑏, 𝑚, and 𝑛 be
positive real numbers
with 𝑏 ≠ 1.
log 𝑏
𝑚
=
𝑛
log 𝑏 𝑚 − log 𝑏 𝑛
Quotient Property of
Exponents
𝑏𝑚
𝑚−𝑛
=
𝑏
𝑏𝑛
“The log of a quotient
equals the difference of
the logs of the divisors.”
Power Property
Power Property of
Logarithms
Let 𝑏, 𝑚, and 𝑛 be
positive real numbers
with 𝑏 ≠ 1.
log 𝑏 𝑚𝑛 =
𝑛 ∙ log 𝑏 𝑚
Power Property of
Exponents
𝑏𝑚
𝑛
= 𝑏 𝑚∙𝑛
“The log of a number to a
power equals the power
times the log of the
number.”
Exercise 2
Expand the following logarithms.
1.
10
ln
9
2. ln 3𝑥 + 2
3.
6𝑥
ln
5
4. ln
2
2
𝑥 +3
3
𝑥 𝑥 2 +1
General Power Rule
Recall that in the General Power Rule, 𝑛 ≠ −1.
𝑛+1
𝑥
𝑥 𝑛 𝑑𝑥 =
+𝐶
𝑛+1
𝑛 ≠ −1
1
𝑑𝑥,
𝑥
If 𝑛 = −1, then we have
which we will use to
formally define the natural logarithm.
The Natural Logarithm
The natural logarithm
is defined by
By this definition, the
natural logarithm is an
accumulation function,
accumulating area under
the graph of
1
𝑦= .
𝑡
𝑥
ln 𝑥 =
1
1
𝑑𝑡
𝑡
𝑥>0
The Natural Logarithm
The natural logarithm
is defined by
By this definition, the
natural logarithm is an
accumulation function,
accumulating area under
the graph of
1
𝑦= .
𝑡
𝑥
ln 𝑥 =
1
1
𝑑𝑡
𝑡
𝑥>0
The Natural Logarithm
The natural logarithm
is defined by
By this definition, the
natural logarithm is an
accumulation function,
accumulating area under
the graph of
1
𝑦= .
𝑡
𝑥
ln 𝑥 =
1
1
𝑑𝑡
𝑡
𝑥>0
The Natural Logarithm
The natural logarithm
is defined by
By this definition, the
natural logarithm is an
accumulation function,
accumulating area under
the graph of
1
𝑦= .
𝑡
𝑥
ln 𝑥 =
1
1
𝑑𝑡
𝑡
𝑥>0
Exercise 3
Use the definition of the natural logarithm to
find ln 1 .
1
ln 1 =
1
1
𝑑𝑡 = 0
𝑡
ln 1 = 0
Graph of 𝑦 = ln 𝑥
The graph of 𝑦 = ln 𝑥 can be derived based on the
1
definition, that ln 𝑥 is an antiderivative of .
𝑥
𝑦 = ln 𝑥
𝑑𝑦 1
=
𝑑𝑥 𝑥
This generates a
slope field with
solution 1,0
Exercise 4
Analyze the graph of 𝑓 𝑥 = ln 𝑥 by
discussing where 𝑓 𝑥 is increasing or
decreasing, its concavity, and any extrema.
Graph of 𝑦 = ln 𝑥
The graph of 𝑦 = ln 𝑥 can be derived based on the
1
definition, that ln 𝑥 is an antiderivative of .
𝑥
Domain: 𝑥 > 0
Range: ℝ
Always Increasing
Concave Down
One-to-One
The Natural Base
For common logs, we know that log 10 = 1,
so there must exist a number such that
ln 𝑥 = 1.
𝑥
ln 𝑥 =
1
1
𝑑𝑡 = 1
𝑡
The area
under 𝑦 = 1/𝑡
is 1
This number is the transcendental number
𝑒 ≈ 2.718281828459045 …
The Definition of 𝑒
The letter 𝑒 denotes the positive real number
such that
𝑒
ln 𝑒 =
1
1
𝑑𝑡 = 1
𝑡
Derivative of ln 𝑥
Let 𝑢 be a differentiable function of 𝑥.
𝑢>0
𝑑
1
ln 𝑥 =
𝑑𝑥
𝑥
𝑑
1 𝑑𝑢 𝑢′
ln 𝑢 = ∙
=
𝑑𝑥
𝑢 𝑑𝑥 𝑢
𝑥>0
Exercise 5
Find each derivative.
1. 𝑦 = ln 2𝑥
2. 𝑦 = ln 𝑥 2 + 1
3. 𝑦 = 𝑥 ∙ ln 𝑥
4. 𝑦 = ln 𝑥 3
Exercise 6
Differentiate 𝑓 𝑥 = ln 𝑥 + 1.
Method 1:
𝑓 𝑥 = ln 𝑥 + 1
Method 2:
1/2
1
𝑓 𝑥 = ln 𝑥 + 1
𝑑
𝑓′ 𝑥 =
𝑥+1
𝑑𝑥
𝑥+1
1
1
𝑓′ 𝑥 =
𝑥+1
𝑥+1 2
𝑓′ 𝑥 =
1
2 𝑥+1
1/2
−1/2
1/2
1
𝑓 𝑥 = ln 𝑥 + 1
2
𝑓′ 𝑥 =
1
2 𝑥+1
Sometimes it’s
convenient to
use properties
of logarithms to
simplify a
problem before
differentiating.
Exercise 7
Find the derivative of 𝑓 𝑥 = ln
𝑥
2
2
𝑥 +1
2𝑥 3 −1
.
Exercise 8
Take the derivative of 𝑦 =
𝑥−2 2
𝑥 2 +1
, 𝑥 ≠ 2, by
first taking the natural log of both sides.
Derivatives with Abs Value
If 𝑢 is a differentiable
function of 𝑥 such
that 𝑢 ≠ 0, then
𝑑
𝑢′
ln 𝑢 =
𝑑𝑥
𝑢
Exercise 9
Find the derivative of 𝑦 = ln cos 𝑥 .
Exercise 10
Locate the relative extrema of 𝑦 = ln 𝑥 2 + 2𝑥 + 3 .
5-1: Natural Logs, 1
Objectives:
1. To define the natural
logarithm and take its
derivative
Assignment:
• P. 329-331: 7-10, 19-33
odd, 37-41 odd, 45, 46,
49-51, 53, 54, 57, 59,
63, 65, 71, 72, 83, 93,
94, 96-98, 103, 104