4.2 Logarithmic FunctionsMAC2233 (3.4) FIU, MDC-North4.2 Logarithmic FunctionsThe inverse function of y = f(x) = ax is x = ay, which is equivalent to the logarithmic function y = log a x. This equivalence is used to convert between logarithmic and exponential functions. Ifthe base is the number e, then x = e y if and only if y = ln x. Properties of common and natural logarithms: (1) logb 1 = 0 , logbb = 1 (b ≠ 1, b > 0), ln1 = 0, and ln e = 1; (2)=blog Mb M, =rblog b r, be bln=, and ln eb = b; and if u and v are any positive numbers, then(3) Equality Rule: = � =b blog u log v u v (4) Product Rule: ( )= +b b blog uv log u log v;(5) Power Rule: =rb blog u r log u; (6) Quotient Rule: = -b b bulog log u log vv; (7) Conversion formulas: bablog Mlog Mlog a and aln Mlog Mln a. Exponential Growth (k > 0) and Decay (k < 0) is given by ( )ktQ t Q e0=, meaning the quantity Q present after t years is equal to the quantity originally present Q0 times ekt where k is a proportionality constant.1. Use your calculator to find (correct to nearest thousandth):a. ln 5 b.1ln5c. ln e-5d.1ln3.5e.e3 4lnf. ln(-5)2. Evaluate using properties of logarithms (no calculator):a.ln e2b.log log6 69 4c.log log5 56 75Simplify, using logarithm rules: 3. log53.5 4.( )-ln x x2 245.xx x43 2ln1� �� �� �-� �Evaluate; round your answer to nearest hundredth.6. log5 18 7.ln 2pSolve for x: 8.x x15 3 ln ln2= -9.xe531 2-=+MAC2233 (3.4) FIU, MDC-North10. How quickly will money double if it is invested at an annual interest rate of 7% compounded continuously? #4011. The editor of a publishing house estimates that if x thousand complimentary copies are distributed to instructors, the first-year sales of the new text will be approximately f(x) = 20 – 12e-0.03x. Approximately how many complimentary copies should the editor send out to generate first-year sales of 12,000 copies? #4412. An economist has compiled these data on GDP for a certain country: 1990: 100 billion, 2000:180 billion.Predict the GDP in 2010 if the GDP is growing: #46a. Linearly:b. Exponentially13. An archaeologist has found a fossil in which the ratio of 14C to 12C is 1/3 the ratio found in the atmosphere. Approximately how old is the fossil. (Half-life of 14C is 5,730 years.) #4814. According to a logistic model based on the assumption that the earth can support no more than 40 billion people, the world’s population (in billions) t years after 1960 is given by the function ( )ktP tCe401-=+, where C and k are positive constants. #64a. Find the function of this form that is consistent with the fact that the world’s population in1960 was 3 billion and in 1975 was 4 billion.b. Predict the population for