Transcript Logarithm – Common and Natural Logarithms

Content Page
 Common Logarithm
 Introduction
 History
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Henry Briggs
Calculators
 Change of Base Law
 Graph
 Natural Logarithm
 Introduction
 History
 Graph
Common Logarithm - Introduction
 Common Logarithms are logarithms to base 10
 Commonly abbreviated as lg
 Hence, for example
Common Logarithm - History
 Sometimes called Briggsian Logarithm
 Named after Henry Briggs, a 17th century
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mathematician
In calculators, when you press log
It is actually log10 or lg
This is because base 10 logarithms are useful for
computations
Engineers often used log to represent log10
Since engineers programmed calculators, log became
log10
Common Logarithm - History
 However, this is extremely misleading
 So we have to take note in case we make such a mistake
by confusing log10 with log
 We often need to make use of logarithms of non-10
bases
 Hence, we will briefly cover the Change of Base Law
Common Logarithm – Change of
Base Law
 If a, b and c are positive numbers and a
1, c
This law is used to manipulate bases, and hence allow us to
overcome to problem of common bases in calculators
1
Common Logarithm – Change of
Base Law
 This law can be used to convert common logarithms to
natural logarithms, and vice versa
 log10N = logeN / loge10
= (ln N) / (ln 10)
= (ln N) / 2.30258
= 0.4343 × ln N
Natural Logarithms- Introduction
• Beside base 10, another important base is e
• where e= 2.71828 (5 d.p)
• Logarithms to base e are called natural logarithms
• “log e”
is often abbreviated as “ln”
Natural Logarithms- Introduction
 Natural logarithms may also be evaluated using the
“ln” button on a scientific calculator.
 By definition,
ln Y = X <-> Y =
x
e
Natural Logarithms- History
 A mathematics teacher, John Speidell, compiled a
table on the natural logarithm in 1619.
 The first mention of the natural logarithm was by
Nicholas Mercator in his work Logarithmotechnia
published year 1668.
 It was formerly known as the hyperbolic logarithm.
Natural Logarithms- Examples
• 1. ln p = 3
2. loge p = 3
3. p = e³
• 1. e2x = k
2. 2x = loge k
3. 2x = ln k