Logarithms 2-1 Definition of Logarithms - PDF document

Logarithms تامتراغوللا 2-1 Definition of Logarithms تامتراغوللا فيرعت If x = b y • then y = log b x where x and b are positive numbers and b ≠ 1 • log b x is read as “logarithm of x to the base of b ”. 2 - 2 Common and Natural Logarithms يعيبطلا متراغوللا و يدايتعلئا متراغوللا • When 10 is the base of a logarithm then it’s called a common logarithm and the base 10 is usually not written. Common logarithm of x = log 10 x = log x • When e is the base of a logarithm then it’s called a natural logarithm and it is usually denoted by the letters ln. Natural logarithm of x = log e x = ln x Where e is a mathematical constant = 2.718281828459045235… • 2 GFP - Sohar University SET 2 - Chapter 2

2 - 3 Laws of Logarithms تامتراغوللا نيناوق • The fundamental laws of logarithms are: SET 2 - Chapter 2 3 GFP - Sohar University Example 1: Change the following from exponential form to logarithmic form: Solution: 4 GFP - Sohar University SET 2 - Chapter 2

Example 2: Change the following from logarithmic form to exponential form: Solution: SET 2 - Chapter 2 5 GFP - Sohar University Example 3: Evaluate the following expressions, rounded to 3 decimal places: Solution: 6 GFP - Sohar University SET 2 - Chapter 2

Example 4: Solve the following logarithmic equations: Solution: SET 2 - Chapter 2 7 GFP - Sohar University Example 5: Solve the following logarithmic equations: Solution: 8 GFP - Sohar University SET 2 - Chapter 2

Example 6: Solve the following exponential equations: Solution: By taking the log 10 for both side: Taking the log 10 for both side gives: SET 2 - Chapter 2 9 GFP - Sohar University Example 7: Express each of the following expressions as a single logarithm: Solution: 10 GFP - Sohar University SET 2 - Chapter 2

2 - 4 Population Growth تاعمتجملا ومن • The model of many kinds of population growth, whether it be a population of people, bacteria, cellular phones, or money is represented by the following function: P ( t ) = P 0 e k t Where: P 0 = the population at time 0, P ( t ) = the population after time t , k = exponential growth rate, and k > 0 SET 2 - Chapter 2 11 GFP - Sohar University Example 8: In 2002, the population of India was about 1034 million and the exponential growth rate was 1.4% per year . (a) Find the exponential growth function. (b) Estimate the population in 2008. (c) After how long will the population be double what it was in 2002? Solution: (a) At t = 0 (2002), the population was 1034 million, then P 0 =1034. k = 1.4%, or 0.014 Therefore, the exponential growth function for this population is: P ( t ) = 1304 e 0.014 t Where t is the number of years after 2002 and P ( t ) is in millions. 12 GFP - Sohar University SET 2 - Chapter 2

(b) In 2008, t = 6 We are looking for the time T for which P ( T ) = 2  1034 = 2068 . (c) To find T , we solve the equation: SET 2 - Chapter 2 13 GFP - Sohar University 2 - 5 Interest Compounded Continuously ةرمتسملا ةبكرملا ةدئافلا • If an amount P 0 is invested in a savings account at interest rate k compounded continuously, then the amount P ( t ) in the account after t years is given by the exponential function: P ( t ) = P 0 e k t where: P 0 = the amount at time 0, P ( t ) = the amount after time t , k = continuously compounded interest rate. 14 GFP - Sohar University SET 2 - Chapter 2

Example 9: Suppose that $2000 is invested at interest rate k , compounded continuously, and grows to $2504.65 in 5 years. (a) What is the interest rate? (b) Find the exponential growth function. (c) What will the balance be after 10 years? (d) After how long will the $2000 be doubled? Solution: (a) P (0) = P 0 = $2000 Thus, the exponential growth function is: P ( t ) = 2000 e kt Since P (5) = $2504.65, Then 2504.65 = 2000 e k (5) 2504.65 = 2000 e 5 k SET 2 - Chapter 2 15 GFP - Sohar University So, the interest rate is 0.045 or 4.5%. Substituting 0.045 for k in the function P ( t ) = 2000 e kt gives: (b) P ( t ) = 2000 e 0.045 t (c) The balance after 10 years is: P (10) = 2000 e 0.045(10) = 2000 e 0.45 = $3136.62 16 GFP - Sohar University SET 2 - Chapter 2

(d) To find the time T needed for doubling the $2000, we set P ( T ) = 2 × P 0 = 2 × $2000 = $4000 and solve for T . 4000 = 2000 e 0.045 T So, the original investment of $2000 will double in about 15.4 years SET 2 - Chapter 2 17 GFP - Sohar University