1 College of ENGINEERING Mathematics I Logarithms Dr Fuad M. Shareef

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

3 Solution Atable can be constructed to show the appreciation of the antique. Time (years) Value Interest It is clear from the table the value exceeds $800 during the fifth year.

4 To find out more precisely when the value is $800. A graph can be constructed to illustrate the growth (nearly 4.9 years) A graph can be constructed to illustrate the growth (nearly 4.9 years) An equation can be found that gives a relationship between time and value. An equation can be found that gives a relationship between time and value. To do this, the method of calculation is analysed more closely: To do this, the method of calculation is analysed more closely:

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

6 After 3 years, t=3 V=500(1.1)3 After 4 years, t=4 V=500(1.1)4 There is clearly a pattern developing. The initial value is multiplied by (1.1) each year. For any value of t years: V=500(1.1)t This equation can be used to find the time when the value reaches $800.

7 500 (1.1)t=800 (1.1)t=800/500 (1.1)t=1.6 What is the value of t?

8 Guesses can be made for time (t) to solve this equation: (1.1)t=1.6 Let t=4 (1.1)4= t=5 (1.1)5= t=4.9 (1.1)4.9= t=4.95 (1.1)4.95= So the required time is about 4.95 years.

9 In an equation such as (1.1) t =1.6, The index (power) t is known as the LOGARITHM of 1.6 to the base 1.1. The index (power) t is known as the LOGARITHM of 1.6 to the base 1.1. This equation can be written as: Log =t In words “ What is the power to which 1.1 must be raised to give 1.6”. In words “ What is the power to which 1.1 must be raised to give 1.6”. Using calculator t= Using calculator t=

10 Summary If N is a positive real number such that N=(b)p exponential form We may write this in alternative form LogbN=p logarithmic form Number = (Base)LOGARITHM

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

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

13 The Laws of Logarithm Log ab = log a+ logb Log ab = log a+ logb Log(a/b) =log a – log b Log(a/b) =log a – log b Log a b = b log a Log a b = b log a Log b a = (log a) / (log b) Log b a = (log a) / (log b) (change of base formula) (change of base formula)