Sect' 10'5 Base e and Natural Logarithms

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Sect. 10.5 Base e and Natural Logarithms
Goal 1 Evaluating Expressions Involving the
Natural Base and Natural Logarithms Goal 2
Solve Exponential Equations and Inequalities
Using Natural Logarithms
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Natural logarithms The system of natural
logarithms has the number called "e" as its base.
 (e is named after the 18th century Swiss
mathematician, Leonhard Euler.)  Base e is used
in theoretical work and is base used in
calculus.  Called the "natural" base because of
technical considerations.  
e is an irrational number, whose value is e ?
2.718281828
To indicate the natural logarithm of a number, we
use the notation "ln."  Thus, ln x    The
logarithm of x with the base e.
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Natural Logarithms

• If x is a positive number, ln x is the exponent
  of e that gives x. That is, y ln x if and
  only if ey x.
• The function ln x is known as the natural
  logarithm function. It may also be written as
  loge x, which is read as log to the base e of
  x.
• Example. What is ln e? Since
  e1 e, it follows that ln e 1.
  What is ln (1/e)? Since e-1 1/e, it
  follows that ln (1/e) -1.
  What is Since
  it follows that

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Natural Logarithms

• Transcendental number
• e 2.71828
• Base of the natural log system
• Exponential function y ex

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• Two important values of the ln function are
• By definition, ln is the inverse function of the
  exponential with base e, so we have
  
• For a and b both positive and any value of t,
  

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Use a Calculator to evaluate each expression to
four decimal places.
a) e0.5
1.6847
b) e- 8
0.0003
c) e5
146.4132
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y ln x
The function defined by f(x) loge x ln x
is called the natural logarithm function.
Use a calculator to evaluate ln 3, ln 2, ln 100
ln 3
LN 3 ENTER
1.0986122
ln 2
ERROR
LN 2 ENTER
ln 100
LN 100 ENTER
4.6051701
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Use a Calculator to evaluate each expression to
four decimal places.
a) ln 3
1.0986
b) ln 0.047
- 3.0576
- 1.3863
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Evaluate each Expression
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a) eln 21
3x 4
b) ln e3x 4
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Write an equivalent exponential or logarithmic
equation.
ln 23 x
a) ex 23
x e1.2528
b) ln x 1.2528
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Properties of Natural Logarithms
1. ln 1 0 since e0 1.
2. ln e 1 since e1 e.
3. ln ex x and eln x x inverse
property
4. If ln x ln y, then x y. one-to-one
property
Examples Simplify each expression.
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Natural logarithms can be condensed/expanded
using the properties of logarithms

• ln 6
• ln x3y
• ln 41/2 2 ln 3
• ln 2 ln 32
• ln 2 ln 9
• ln 18

Condense the expressionsa. ln 18 ln
3b. 3ln x ln y c.
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Solve
ln 3x 0.5
ln 3x ln e0.5
3x e0.5
3x 1.6487
x 0.5496
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Solve
2ex 5 1
2ex 6
ex 3
ln ex ln 3
X 1.0986
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Solve
ln (2x 3) lt 2.5
ln (2x 3) lt ln e2.5
Make sure the logarithm is greater than 0
0 lt 2x 3 lt e2.5
0 lt 2x 3 lt 12.18249
3 lt 2x lt 15.18249
1.5 lt x lt 7.5912
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Solve
3 e- 2x 8
e- 2x 5
ln e- 2x ln 5
- 2x 1.60943
x - 0.8047
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Natural Logarithms
In the study of exponents and logs, different
bases are often used. It would simplify matters
to use the same base every time. Mathematicians
have found that using base e is very advantageous.
In 1987, the worlds population was 5 billion
and was increasing at a rate of 1.6 per annum.
An equation for this, as a function of time is
P 5(1.016)t
Express this using base e
Let 1.016 ek.
By definition, k ln 1.016.
ln 1.016 0.0158733
0.016 (to 2 significant digits)
Hence, 1.016 e0.016.
The equation can now be written as
The exponent of base e is the growth rate.
P 5e0.016t
Growth Rate
Initial population
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When interest is compounded continuously, the
amount A in an account after t years with a
Principal of P and annual interest rate of r is
found using the formula A Pert
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Suppose you deposit 700 into an account paying
6 annual interest, compounded continuously.
b) How long will it take for the balance in your
account to reach 2000?
a) What is your balance after 8 years?
A Pert
A Pert
A 700e(.06)(8)
2000 700e(.06)t
A 700e.48
2.85714 e(.06)t
A 700(1.61607)
ln 2.85714 ln e(.06)t
A 1131.25
1.0498 .06t
17.4970 t
At least 17.5 years