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College of ENGINEERING College of ENGINEERING Mathematics IMathematics I

Logarithms Logarithms

Dr Fuad M. ShareefDr Fuad M. Shareef

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Why Logarithms?Why Logarithms?Problem:Problem:An antique is valued at $500 and An antique is valued at $500 and appreciates at 10% per annum. How may appreciates at 10% per annum. How may years will it takes before the value exceeds years will it takes before the value exceeds $800? $800?

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SolutionSolution A table can be constructed to show the A table can be constructed to show the

appreciation of the antique.appreciation of the antique.Time (years) Value InterestTime (years) Value Interest 0 500.00 50.000 500.00 50.00 1 550.00 55.001 550.00 55.00 2 605.00 60.002 605.00 60.00 3 665.50 66.553 665.50 66.55 4 732.05 73.214 732.05 73.21 55 805.26 805.26 It is clear from the table the value exceeds It is clear from the table the value exceeds

$800 during the fifth year.$800 during the fifth year.

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To find out more precisely when the To find out more precisely when the value is $800.value is $800. A graph can be constructed to illustrate A graph can be constructed to illustrate

the growth (nearly 4.9 years)the growth (nearly 4.9 years) An equation can be found that gives a An equation can be found that gives a

relationship between time and value.relationship between time and value.

To do this, the method of calculation is To do this, the method of calculation is analysed more closely:analysed more closely:

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Initial time, t=0 Value, V=500Initial time, t=0 Value, V=500After 1 year, t=1 V=500+10%of 500After 1 year, t=1 V=500+10%of 500 V=500+0.1x500V=500+0.1x500 V=500(1+0.1)V=500(1+0.1) V=500(1.1)V=500(1.1)When t=2, When t=2, V=500(1.1)+10% of 500(1.1)V=500(1.1)+10% of 500(1.1) V=500(1.1)+0.1x500(1.1)V=500(1.1)+0.1x500(1.1) V=500(1.1)[1+0.1]V=500(1.1)[1+0.1] V=500(1.1)(1.1)V=500(1.1)(1.1)

That is after 2 years V=500(1.1)That is after 2 years V=500(1.1)22

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After 3 years, t=3 V=500(1.1)After 3 years, t=3 V=500(1.1)33

After 4 years, t=4 V=500(1.1)After 4 years, t=4 V=500(1.1)44

There is clearly a pattern developing.There is clearly a pattern developing. The initial value is multiplied by (1.1) The initial value is multiplied by (1.1)

each year.each year. For any value of t years:For any value of t years: V=500(1.1)V=500(1.1)tt

This equation can be used to find the This equation can be used to find the time when the value reaches $800.time when the value reaches $800.

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500 (1.1)500 (1.1)tt=800=800(1.1)(1.1)tt=800/500=800/500

(1.1)(1.1)tt=1.6=1.6What is the What is the value of value of tt??

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Guesses can be made for time Guesses can be made for time (t) to solve this equation:(t) to solve this equation:

(1.1)(1.1)tt=1.6=1.6Let t=4 (1.1)Let t=4 (1.1)44=1.4641 =1.4641 t=5 (1.1)t=5 (1.1)55==1.610511.61051 t=4.9 (1.1)t=4.9 (1.1)4.94.9=1.5952331=1.5952331 t=4.95 (1.1)t=4.95 (1.1)4.954.95==1.61.6028534028534So the required time is about So the required time is about

4.954.95 years. years.

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In an equation such as (1.1)In an equation such as (1.1)tt=1.6,=1.6, The index (power) t is known as the The index (power) t is known as the

LOGARITHM LOGARITHM of 1.6 to the base 1.1.of 1.6 to the base 1.1.This equation can be written as:This equation can be written as:

LogLog1.11.11.6=t1.6=t In words “ In words “ What is the power to which What is the power to which

1.1 must be raised to give 1.61.1 must be raised to give 1.6”. ”.

Using calculator t= Using calculator t= log1.6 4.9log1.1

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SummarySummary

If N is a positive real number such If N is a positive real number such that N=(b)that N=(b)p p exponential formexponential form

We may write this in alternative We may write this in alternative formform

LogLogbbN=p N=p logarithmic formlogarithmic form

Number = (Base)Number = (Base)LOGARITHMLOGARITHM

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Common Logarithms Common Logarithms

When the number 10 used as When the number 10 used as the base for logarithm, then the base for logarithm, then they are called common they are called common logarithm and is denoted by logarithm and is denoted by LogLog10 10 ..

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Natural logarithmsNatural logarithms

When the number When the number e=2.718281828… used as the e=2.718281828… used as the base for logarithm, then they base for logarithm, then they are called Natural logarithm are called Natural logarithm and is denoted by ln N . and is denoted by ln N .

ln N means Logln N means LogeeN.N.

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The Laws of LogarithmThe Laws of Logarithm

Log ab = log a+ logbLog ab = log a+ logb Log(a/b) =log a – log bLog(a/b) =log a – log b Log aLog abb = b log a = b log a LogLogbba = (log a) / (log b)a = (log a) / (log b) (change of base formula)(change of base formula)