advertisement

Using logarithms to solve for t in exponential growth problems 1 Logarithms • Use exponents as an alternative way to represent numbers 2 Logarithms • Use exponents as an alternative way to represent numbers • Tell the power to which the base number (usually 10) must be raised to obtain a value X 3 Logarithms • Alternative way of representing numbers using exponents • Tell the power to which the base number (usually 10) must be raised to obtain a value X • log10X signifies the power to which 10 must be raised to equal X 4 Logarithms y log10X = Y is identical to 10 = X 5 An example of logarithms The log of 100 in base 10 log10100 = 2 because 2 10 =100 6 Another example of logarithms The log of 1000 in base 10 [Log101000] = 3 because 3 10 =100 7 A third example The log of 30 in base 10 [Log1030] = 1.477 because 1.477 10 =30 8 Calculate logs in Excel using Log functions =Log(number) 9 In Excel, compute 1. The log of your age 2. The log of your weight 3. The log of your number of credits [If you are a first-term freshman, use the number of credits you will have after this quarter] 10 Key rule for using logarithms: x Log a = x * log a 11 An example of the use of logs A Petri dish contains 100 bacteria cells. The number of cells increases 5% every minute. How many minutes will it take for there to be 3000 bacteria in the dish? 12 An example of the use of logs Using the simulation method, we can find that there will be 2898 cells in minute 69 and 3043 in minute 70 but it will not tell us exactly when there were 3000 cells. 13 The example continued Begin with the formula for exponential growth: t 3000 = 100 * 1.05 That is, at 5% growth per minute, 100 cells will grow into 3000 in t minutes. 14 The example continued Step 2: divide both sides by 100, which gives us: t 30 = 1.05 15 The example continued Step 3: take the log of each side: t Log(30) = log(1.05 ) 16 The example continued t Step 4: apply the rule [log a = t*log a] log 30 = t* log 1.05 17 The example continued Step 5, divide both sides by log 1.05: log 30/log 1.05 = x 18 The example completed Step 6, use Excel to compute the value of x. =log(30)/log(1.05) 69.71 = x 19