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Logarithmic Equations log ๐ ๐ = ๐ฅ is equivalent to ๐ = ๐ ๐ฅ The default base for logarithms is 10 log10 100 Example: log 2 8 = 3, because 8 = 23 Example: log 100 is the same as The natural log is ln ๐ฅ, which has base ๐ ๐ โ 2.718281828 โฆ ln ๐ฅ = log ๐ ๐ฅ Properties of Logarithmic Equations log ๐ 1 = 0 log ๐ฅ๐ฆ= log ๐ฅ + log ๐ฆ log ๐ ๐ = 1 log ๐ฆ = log ๐ฅโ log ๐ฆ ๐ฅ ln ๐ = 1 log ๐ ๐x =x log ๐ฅ ๐ฆ = log ๐ฆ log ๐ฅ ln ๐ ๐ฅ = ๐ฅ If log ๐ฅ = log ๐ฆ, then ๐ฅ = ๐ฆ log ๐ ๐x = x log ๐ ๐ For log ๐ ๐ฅ, the domain is x >0, otherwise the logarithm is undefined ๐log๐ ๐ = b Solving an Exponential or Logarithmic Equation Solve the equation: 4๐ฅ = 87 log 4๐ฅ = log 87 *Take the log on both sides x log 4 = log 87 *Move the exponent (x) to the front of the log x= log 87 log 4 โ 3.221 *Divide both sides by log 4 Solve the equation: log 5 ๐ฅ = 6 ๐ฅ = 56 *Rearrange the equation so that the log is not needed anymore ๐ฅ = 15625 Expand the following expression and simplify: ln 15๐๐ 3 ln 15๐๐ 3 = ln 15 + ln ๐ + ln ๐ 3 *Expand the expression ln 15 + ln ๐ + ln ๐ 3 = ln 15 + ln ๐ + 3 ln ๐ *Move the exponent (3) to the front of the natural log The final answer is ln 15 + ln ๐ + 3 ln ๐ Compound Interest Questions ๐ General Compound Interest Equation: ๐ด = ๐(1 + ๐)๐๐ก Compounded Continuously: ๐ด = ๐๐ ๐๐ก A = Amount at the end of the time period P = Principal amount put in r = Interest rate (as a decimal) n = Number of times compounded per year t = Length of the time period in years ๐ โ 2.718281828 โฆ โ ๐๐ ๐ ๐กโ๐ ๐ ๐ฅ ๐๐ข๐ก๐ก๐๐ ๐๐ ๐ฆ๐๐ข๐ ๐๐๐๐๐ข๐๐๐ก๐๐ Examples: $12oo is put into a bank account that earns 8% interest. Find the amount of money in the bank account after 5 years if the interest is compounded annually, monthly, and continuously. Round to the nearest cent. In this particular problem: A = Amount in the account after the 5 years (also what we are trying to find) P = $1200 r = 8% = .08 t = 5 years n = 1 for annually n = 12 for monthly ๐ Annually: ๐ด = ๐(1 + ๐)๐๐ก ๐ด = 1200(1 + .08 1โ5 ) 1 ๐ด = 1200(1.08)5 ๐ด = $ 1763.19 *Make sure the 1.08 is in parenthesis in your calculator so that the correct order of operations is followed ๐ Monthly: ๐ด = ๐(1 + ๐)๐๐ก ๐ด = 1200(1 + . 08 12โ5 ) 12 ๐ด = 1200(1.006ฬ )60 ๐ด = $ 1787.81 Continuously: ๐ด = ๐๐ ๐๐ก ๐ด = 1200๐ .08โ5 ๐ด = 1200๐ .4 ๐ด = $ 1790.19 You are trying to save $80,000 to help you purchase a home. You start with $25,000 in a bank account that gains 4% interest. How long will it take you to save the $80,000 you need if the interest is compounded annually, quarterly, and continuously? Round to the nearest hundredth. In this particular problem: A = $80,000 P = $25,000 r = 5% = .05 t = How long it will take you save the money you need (what we are trying to find) n = 1 for annually n = 4 for quarterly ๐ Annually: ๐ด = ๐(1 + ๐)๐๐ก 80000 = 25000(1 + 80000 25000 .05 1โ๐ก ) 1 = (1 + .05)๐ก *Divide by 25000 on both sides 3.2 = (1.05)๐ก *Add inside the parenthesis log 3.2 = log 1.05๐ก *Take the log on both sides log 3.2 = ๐ก log 1.05 *Move the exponent (t) to the front of the log log 3.2 log 1.05 *Divide by log 1.05 on each side =๐ก ๐ก = 23.84 ๐ฆ๐๐๐๐ ๐ Quarterly: ๐ด = ๐(1 + ๐)๐๐ก 80000 = 25000(1 + 80000 = (1 + 25000 .05 4๐ก ) 4 .05 4โ๐ก ) 4 *Divide by 25000 on both sides 3.2 = (1.0125)4๐ก *Add inside the parenthesis log 3.2 = log 1.01254๐ก *Take the log on both sides log 3.2 = 4๐ก log 1.0125 *Move the exponent (4t) to the front of the log log 3.2 log 1.0125 *Divide by log 1.0125 on each side = 4๐ก 4๐ก = 93.6324 ๐ก = 23.41 ๐ฆ๐๐๐๐ Continuously: ๐ด = ๐๐ ๐๐ก 80000 = 25000๐ .05โ๐ก 80000 25000 = ๐ .05๐ก *Divide by 25000 on both sides 3.2 = ๐ .05๐ก ln 3.2 = ln ๐ .05๐ก *Take the natural log on both sides (ln goes with e) ln 3.2 = .05๐ก ln ๐ *Move the exponent (.05t) to the front of the log ln 3.2 = .05๐ก *ln ๐ = 1 ๐ก= ln 3.2 .05 ๐ก = 23.26 ๐ฆ๐๐๐๐ *Divide by .05 on both sides