Why Should We Study LogarithmsIn exams, since some of them do not allow the use of calculators, logarithms are used to calculate amounts in compound interest. The study of logarithms arose mainly as an aid in solving exponential equations. It is also present in mathematical models used in several areas.In other words, logarithms can transform exponential equations into linear equations! That is where its usefulness comes from.> Given a and b positive real numbers, the logarithm of b in base a is called the exponent to which a must be raised so that the resulting power with base a is equal to b.\[log_a b = x <=> a^{x} = b\]Thus, a logarithm is nothing more than an exponent. We say that "a" is the base of the logarithm, "b" is the argument of the logarithm, and "x" is the logarithm.Logarithm PropertiesI) The logarithm whose argument is equal to 1 and whose base is any number is equal to zero;II) The logarithm whose base and argument are equal is equal to one;III) Two logarithms are equal, in the same base, if their arguments are equal;IV) Logarithm of a product:\[log_c(a · b) = log_c(a) + log_c(b)\]V) Logarithm of a division:\[log_c(\frac{a}{b}) = log_c(a) - log_c(b)\]VI) Logarithm of a power:\[log_c(a^{b}) = b · log_c(a)\]VII) Logarithm of a root:\[log_c(a^{\frac{1}{b}}) = \frac{log_c(a)}{b}\]Logarithms in Financial Mathematics and ExamsStarting from the already known compound interest equation:\[ M_t = C · (1 + i)^{t} \]\[ log(M_t) = log(C · (1 + i)^{t}) \]\[ log(M_t) = log(C) + log((1 + i)^{t}) \]\[ log(M_t) = log(C) + t · log(1 + i) \]It is noticeable that the previously exponential equation becomes linear. In other words, every time time increases by 1, log(1 + i) is added to the principal, always adding the same amount no matter the moment.ExamplesAn amount of R$ 10,000.00 was invested at compound interest of 2% per month for 125 months. Knowing that log(1.02) = 0.008, what is the amount of this operation?\[ log(M_t) = log(C) + t · log(1 + i) \]\[ log(M_t) = log(10.000) + 125 · log(1 + 0,02) \]\[ log(M_t) = 4 + 125 · 0,008 \]\[ log(M_t) = 4 + 125 · 0,008 = 5 \]\[ M_t = 100.000 \]Júlio has an amount Q, in reais, and intends to invest it, under compound interest, at a rate of 4% per month. Considering log 2 = 0.3010 and log 1.04 = 0.0086, how long will it take for this amount to quadruple?\[ log(M_t) = log(C) + t · log(1 + i) \]\[ log(4 · C) = log(C) + t · log(1 + 0,04) \]\[ log(4 · C) - log(C) = t · log(1,04) \]\[ log(\frac{4 · C}{C}) = t · log(1,04) \]\[ log(\frac{4}{1}) = t · log(1,04) \]\[ log(2^{2}) = t · 0,0086 \]\[ 2 · log(2) = t · 0,0086 \]\[ 2 · 0,3010 = t · 0,0086 \]\[ t = \frac{2 · 0,3010}{0,0086} = \text{5 years and 10 months} \]Term CalculationWe want to know the term of a transaction, using only interest, amount, and principal:- Interest: 1% every 30 days;- Amount: R$ 10,803.46;- Principal: R$ 10,000.00;\[ M = C · (1 + i)^{\frac{n}{m}}\]\[ \frac{M}{C} = (1 + i)^{\frac{n}{m}}\]\[ ln(\frac{M}{C}) = \frac{n}{m} · ln(1 + i)\]\[ n = m · \frac{ln(\frac{M}{C})}{ln(1 + i)}\]The exact number of days between the loan date and the receipt date is 233.00 days, obtained with:\[ n = 30 · \frac{ln(\frac{10.803,46}{10.000})}{ln(1 + 0,01)} = 233,0014\]Exercises1) An investment yields compound interest at a rate of 6% per year. After how many years will an initial amount of R$ 1,000.00 reach the amount of R$ 10,000.00 with this investment? (Use log(1.06) = 0.025 )2) An investor bought a lot of shares of a company for R$ 1000.00 and resold it, after n months, for R$ 4000.00. Assuming that the monthly appreciation of these shares was 8% per month and using the approximations log 2 = 0.3 and log 3 = 0.48, the value of n is?3) Saulo invested R$ 45,000.00 in an investment fund that yields 20% per year. His goal is to use the amount from this investment to buy a house that, on the date of the investment, cost R$ 135,000.00 and appreciates at an annual rate of 8%. Under these conditions, from the date of the investment, how many years will pass until Saulo can buy such a house? Given: (Use the approximation: log 3 = 0.48)4) (PUC-SP) A principal C, invested at compound interest at a rate i per period, produces, at the end of n periods, the amount M, given by M = C . (1+ i)n. Under these conditions, using Log 2 = 0.30 and log 3 = 0.48, the principal of R$ 2,000.00, invested at compound interest at a rate of 20% per year, will produce the amount R$ 5,000.00 at the end of a period of?5) (CESGRANRIO-2008) An investment of R$1,000.00 was made under compound interest at a rate of 3% per month. After a period t, in months, the amount was R$1,159.27. What is the value of t? (Data: ln(1.000) = 6.91; ln(1.159,27) = 7.06 ; ln(1.03) = 0.03)6) (ESPM) An amount of R$ 10,000.00 was invested at compound interest of 4% per month for 10 months. Knowing that log 1.04 = 0.017 and log 1.48 = 0.17, we can conclude that the interest obtained in this investment was?7) (You may use a calculator) To settle the financing of R$175,000, the amount paid was R$197,338.77. Considering an interest rate of 2% every 30 days, calculate the term of this financing.8) (You may use a calculator) In what time, for each R$1 deposited in an account remunerated by the performance of a stock portfolio, will R$2.12023 be redeemed considering that the return history of this portfolio is 4.20% every 30 days?9) (You may use a calculator) Calculate how much time a capital can take to generate 40% interest if it is invested at an interest rate of 35.02% over 365 days.Answer Key1) 40 years.2) 15 months.3) 12 years.4) 5 years.5) t = 5.6) R$ 4,800.007) 181 days.8) 547 days.9) 409 days.
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