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Understanding Logarithms and Exponents in Financial Calculations

This guide explores the properties of logarithms and exponents, essential for solving equations in financial contexts. It covers continuous and discrete compounding of interest, demonstrating how to calculate total repayments for loans, tuition fees inflation, and depreciation of assets like vehicles. With practical examples including Sparky's loan and tuition costs, readers will learn to model financial scenarios mathematically, using equations to predict future values and assess growth or decline over time.

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Understanding Logarithms and Exponents in Financial Calculations

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Presentation Transcript

  1. Final Review

  2. Use the properties of logarithms and exponents to solve for x. • The following inverse properties hold for logarithms with base a.

  3. Use the properties of logarithms and exponents to solve for x. • The following inverse properties hold for logarithms with base a.

  4. Use the properties of logarithms and exponents to solve for x. • The following inverse properties hold for logarithms with base a.

  5. Use the properties of logarithms and exponents to solve for x. • The following inverse properties hold for logarithms with base a.

  6. If a principal of P dollars is deposited in an account paying an annual rate of interest r (expressed in decimal form), compounded continuously, then after t years the account will contain A dollars, where A = Pert If a principal P dollars is deposited in an account paying an annual rate of interest r (expressed in decimal form), compounded (paid) n times per year, then after tyears the account will contain A dollars, where Sparky borrowed a loan of $8000 at an interest rate of 7% which is compounded quarterly. He has to pay the loan in 2 years. What is the total amount of money William has to pay? Compounded Quarterly: Let P = 8000, r = 0.07, n = 4, and t = 2 Compounded Continuously: Let P = 8000, r = 0.07, and t = 2

  7. The cost of tuition at a typical 4-year college is currently $7500. This cost is inflating at a rate of 5.6 % per year. Write an equation, f(x), that will model this situation. To the nearest dollar, what will be the cost of tuition in 4 years? To the nearest tenth of a year, how long it will take for the cost of tuition to double?

  8. Sparky bought a used go cart for $4000. The go cart will depreciate at 4.5% per year. Write an equation, f(x), that will model this situation. To the nearest dollar, how much will the go cart be worth in 4 years?

  9. How many terms are there in the following sequence? This is an arithmetic sequence with It is given by the formula:

  10. Find the following sum: The sum of the infinite geometric sequence with first term a1 and common ratio r isgiven by

  11. Find the following sum: The sum of a finitearithmetic sequence, denoted Sn,is found by averaging the first and nth terms and then multiplying by n. That is,

  12. Change of Base Formula Let x, a ≠ 1, and b ≠ 1 be positive real numbers. Then Use a calculator to approximate

  13. Properties of the Logarithm For positive numbers m, n, and a ≠ 1 and any real number r :

  14. Properties of the Logarithm For positive numbers m, n, and a ≠ 1 and any real number r :

  15. Evaluating Combined Functions (fg)(0) is not defined, since 0 is not in the domain of f(x).

  16. Evaluating Composition of Functions numerically

  17. Building an inverse step by step (example) List the operations for f List the opposite operations for f inverse in the opposite order Build f inverse Building the inverse

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