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Application of LogarithmThe following famous saying of mathematical Laplace gives crux of application of logarithm in mathematics. "The invention of logarithms shortens the calculations extending over months to just a few days and thereby as it were doubles the life span of calculator."Now we will see how it shortens the calculations. Examples on application of logarithm1) If $1750 is invested at 9% interest per year for 10 years. Find a) the interest compounded annually. b) the interest compounded half yearly and , c) the difference between (a) and (b).Solution : Using the formula, A = P(1 + r) ^{t} ∴ A = 1750 (1 + 0.09) ^{10} log A = log(1750) + 10 log(1.09) = 3.2430 + 10 x 0.03740 = 3.2430 + 0.3740 log A = 3.6170 ∴ A = 4139.996 = $ 4140 (a) Interest = 4140  1750 = $2390 (b) the interest compounded half yearly A = P(1 + r/2) ^{2t} ∴ A = 1750 (1 + 0.09/2) ^{20} ∴ A = 1750 (1 + 0.045) ^{20} log A = log(1750) + 10 log(1.045) = 3.2430 + 20 x 0.01911 = 3.2430 + 0.382325 log A = 3.6250 ∴ A = $4216.96 = $ 4217 (a) Interest = 4217  1750 = $2467 (c) Difference between (a) and (b) = $2467  $2390 = $77 ∴ The interest compounded half yearly is $77 more that the interest compounded annually. 2) The initial bacterium count in a culture is 200. A biologist later makes a sample count of bacteria in the culture and finds that the relative rate of growth is 30% per hour. (a) Find a function that models the number of bacteria after t hours. (b) What is the estimated count after 1 hour? (c) What is the estimated count after 8 hours? Solution : We will use n _{0} as initial population, n is the population after time t in hours.Rate = r (in percent) Formula : n(t) = n_{0} e^{rt} (a) n(t) = 200 e ^{0.3t} Since here we have used a exponential growth so we will use natural logarithm. (b) The estimated count after 1 hour n(1) = ln (200) + 0.3 ln(e) (For t = 1 hour ) n(1) = 5.29831 + 0.3(since ln(e) = 1) n(1) = 5.59831 n(1) = 269.96 = 270 (c) The estimated count after 8 hours n(8) = ln (200) + 2.4 ln(e) (since t = 8 hour ) n(8) = 5.29831 + 2.4(since ln(e) = 1) n(8) = 7.69831 n(8) = 2204.619 = 2204.62 Application of logarithm to Home page Covid19 has led the world to go through a phenomenal transition . Elearning is the future today. Stay Home , Stay Safe and keep learning!!! Covid19 has affected physical interactions between people. Don't let it affect your learning.
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