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Logarithm – Common and Natural Logarithms

Logarithm – Common and Natural Logarithms. Content Page. Common Logarithm Introduction History Henry Briggs Calculators Change of Base Law Graph Natural Logarithm Introduction History Graph. Common Logarithm - Introduction. Common Logarithms are logarithms to base 10

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Logarithm – Common and Natural Logarithms

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  1. Logarithm – Common and Natural Logarithms

  2. Content Page • Common Logarithm • Introduction • History • Henry Briggs • Calculators • Change of Base Law • Graph • Natural Logarithm • Introduction • History • Graph

  3. Common Logarithm - Introduction • Common Logarithms are logarithms to base 10 • Commonly abbreviated as lg • Hence, for example

  4. Common Logarithm - History • Sometimes called Briggsian Logarithm • Named after Henry Briggs, a 17th century 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

  5. 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

  6. Common Logarithm – Change of Base Law • If a, b and c are positive numbers and a 1, c 1 This law is used to manipulate bases, and hence allow us to overcome to problem of common bases in calculators

  7. 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

  8. 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”

  9. Natural Logarithms- Introduction • Natural logarithms may also be evaluated using the “ln” button on a scientific calculator. • By definition, ln Y = X <-> Y = ex

  10. 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.

  11. 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

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