Core 3 Natural Logarithms

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Core 3 Natural Logarithms

• Learning Objective
• Understand what a natural logarithm is.

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Logs (1)

• Laws of logs
• log a log b log ab
• Example
• log 3 log 5
• log 15
• log a - log b log (a/b)
• Example
• log 21 log 7
• log 3

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Logs (2)

• Laws of logs
• log a k k log a
• Example
• log 35
• 5 log 3

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e 2.718281828459045... the Base of Natural
Logarithms
The number e was first studied by the Swiss
mathematician Leonhard Euler in the 1720s,
although its existence was more or less implied
in the work of John Napier, the inventor of
logarithms, in 1614. Euler was also the first to
use the letter e for it in 1727 (the fact that it
is the first letter of his surname is
coincidental). As a result, sometimes e is called
the Euler Number, the Eulerian Number, or
Napier's Constant (but not Euler's Constant).
An effective way to calculate the value of e is
to use the following infinite sum e 1/0!
1/1! 1/2! 1/3! 1/4! .
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Why e ?
e is a real number constant that appears in some
kinds of mathematics problems. Examples of such
problems are those involving growth or decay
(including compound interest), the statistical
"bell curve," the shape of a hanging cable (or
the Gateway Arch in St. Louis), some problems of
probability, some counting problems, and even the
study of the distribution of prime numbers. It
appears in Stirling's Formula for approximating
factorials. It also shows up in calculus quite
often, wherever you are dealing with either
logarithmic or exponential functions. There is
also a connection between e and complex numbers,
via Euler's Equation.
Also, youll soon see, its unique properties
make it very powerful in calculus
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y ex
When x0 y e0 1
So it goes through (0,1)
The x-axis is an asymptote
y ex can, of course, be transformed
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Natural Logarithms

• Any log to the base e is known as a
• natural logarithm.
• In French this is a
• logarithme naturel
• Which is where ln comes from.
• When you see ln (instead of log)
• then its a natural log

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The laws of logs still hold

• Laws of logs
• ln a ln b ln ab
• Example
• ln 2 ln 8 ln 16
• ln a - ln b ln (a/b)
• Example
• ln 42 ln 6 ln 7

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and for these

• Laws of logs
• ln a k k ln a
• Example
• ln 35 5 ln 3

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Natural Logarithms eg1

• Solve
• ex 7
• can use trial and improvement
• x1.95
• or use the Laws of Logs

ln ex ln 7
x ln e ln 7
x ln 7 x 1.9459
Which is quicker and more accurate
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Natural Logarithms eg2

• Solve
• 2 e3x 7
• use the Laws of Logs

e3x 7/2 3.5
3x ln e ln 7/2
3x ln 7/2 3x 1.2528
x 0.418
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Natural Logarithms eg3

• Solve
• e3x2 - 1 7
• use the Laws of Logs

e3x2 7 1 8
(3x2) ln e ln 8
3x2 ln 8 3x2 2.079
3x 2.079 2 0.079
x 0.079/3 0.026
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Exercise

• Complete the idea development B questions on page
  74 and 75.
• B2 to B13

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Homework

• Core 3 EXERCISE B QUESTIONS 1-3