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7/29/2020 B02005 – Chapter 0: Introduction
CHAPTER 2:
EXPONENTS AND
LOGARITHMS
COURSE CODE: B
1
PREPARED BY: FINANCE DEPARTMENT
LEARNING OBJECTIVES
Introduce the present value. Define the form of the annuities. Identify and explain the optimal holding time. Introduce the logarithmic derivative. 7/29/2020 B03013^ –^ Chapter^ 2: Exponents and Logarithms 2
CONTENT
- Present value 2
- Annuities 3
- Optimal holding time 4
- Logarithmic Derivative. 7/29/2020 B03013^ –^ Chapter 2: Exponents and Logarithms 3
CONCEPTS
- Exponent and exponential function.
- Logarithm and logarithmic function 7/29/2020 B03013^ –^ Chapter 2: Exponents and Logarithms 4
Logarithm
- To solve mathematical expressions that include exponents.
- Ex: 10x=
- Could be iterated as “To what power must the number 10 be raised to equal 100?”. 7/29/2020 B03013^ –^ Chapter 2: Exponents and Logarithms 7
Logarithm
- But in logarithms: + 1st^ : to solve the question as a logarithm. + Then : solve for x using a logarithm table.
- The relationship 10 x=100 can be expressed using logarith notation: x=log 10 100=
- => « log base 10 of 100 is 2 ». 7/29/2020 B03013^ –^ Chapter 2: Exponents and Logarithms 8
Logarithm
- Logarithmic derivative :
- The derivative of the logarithmic function y =lnx
- Derivative of y = ln u (where u is a function of x ) 7/29/2020 B03013^ –^ Chapter 2: Exponents and Logarithms 9
Logarithm
- Example Y= 𝑥^2 − 1 4 𝑥^2 + 1 To compute the derivative 7/29/2020 B03013^ – and Logarithms^ Chapter 2: Exponents 10
Applications
- Annuities: 7/29/2020 B03013^ –^ Chapter 2: Exponents and Logarithms 13
Applications
Example: You have started your first job and decide to put $200 a month into an annuity. The annuity earns 7.2% interest per year, compounded monthly. How long (in months and years) will it take for the account to be worth $1,000,000? 7/29/2020 B03007^ –^ Chapter 2: Exponents and Logarithms 14
Applications
7/29/2020 B03007^ –^ Chapter 2: Exponents and Logarithms 15
Applications
- Optimal holding time : a decision to hold an asset in a certain period.
- There are many assets: appreciate/depreciate over time- vintage wine, real estate, forestry plantation and mining to name a few.
- Compare an investment to others by the present value. 7/29/2020 B03007^ – and Logarithms^ Chapter 2: Exponents 16
Applications
- Example 3 : You have 500 USD, you sent it in your bank account. It the interest rate is 8% per year. How much your account will increase to after 5 years? 7/29/2020 B03007^ – and Logarithms^ Chapter 2: Exponents 19
Exercise
- At 10 percent annual interest rate, which of the following has the largest present value a) $215 two years from now b) $100 after each of the next two years, or c) $100 now and $95 two years from now
- Assuming a 10 percent interest rate compounded continuously, what is the present value of an annuity that pay $500 a year a) For next five years b) Forever? 7/29/2020 B03007^ – and Logarithms^ Chapter 2: Exponents 20
Exercise
3) Suppose you own a rare book whose value at time t years from now will be B(t) = 2 √𝑡^ dollars. Assuming a constant interest rate of 5%, when is the best time to sell the book and invest the proceeds 7/29/2020 B03007^ – and Logarithms^ Chapter 2: Exponents 21
Summary
- Reading the law of exponents and logarithmic.
- Applying the forms of those functions in economics (interest compounding, optimal holding time…).
- Your discussions: how can we calculate some macro-factors in Vietnam. 7/29/2020 B03007^ –^ Chapter 2: Exponents and Logarithms 22
