Properties of Logarithms The Product Rule for Exponents - b m b n = b m+n Example Expand the following: a) log 4 (7 5) log b MN = log b M + log b N b) log (10x) The Power Rule for Exponents - (b m ) n = b mn Example Expand the following: a) log 5 7 4 log b M p = p log b M b) ln x c) log (4x) 5 1 Page Province Mathematics Southwest TN Community College
The Quotient Rule for Exponents: bm bn = bm n log b M N = log bm log b N Example Expand the following: a) log 7 19 x b) ln e3 7 Example Compound Examples: Expand the following a) log b x2 y 5 z 4 3 b) log b a2 b c 5 2 Page Province Mathematics Southwest TN Community College
Example Express as a Single Logarithm a) log b x log b y + 1 4 log bz b) log 4 2 + log 4 32 c) log(4x 3) logx 3 Page Province Mathematics Southwest TN Community College
The Change of Base Property For any logarithmic bases a and b, and any positive number M, log b M = logm logb The logarithm of M with base b is equal to the logarithm of M with any new base divided by the logarithm of b with that new base. Example Use change of base for the following Change to base 10 and to base e a) log 5 8 b) log 7 2506 4 Page Province Mathematics Southwest TN Community College
Exponential and Logarithmic Equations Base Exponent Property For any b>0 and b 1 b x = b y x = y Example Solve for the variable a) 2 5x = 64 d) 27 x+3 = 9 x 1 b) 2 3x 7 = 32 e) 5 3x 6 = 125 c) 8 x+2 = 4 x 3 f) 4 x = 15 5 Page Province Mathematics Southwest TN Community College
g) 10 x = 120,000 h) 40e 0.6x 3 = 237 i) 5 x 2 = 4 2x+3 6 Page Province Mathematics Southwest TN Community College
Recall: log b M = c means b c = M Example Solve for the variable a) log 4 (x + 3) = 2 b) 3ln (2x) = 12 One to One Property of Logarithms For any M>0, N>0, b>0, and b 1 log b M = log b N M = N Example Solve for the variable a) ln(x + 2) ln (4x + 3) = ln 1 x 7 Page
b) ln(x 3) = ln(7x 23) ln (x + 1) Example Medical Research indicates that the risk of having a car accident increases exponentially as the concentration of alcohol in the blood increases. The risk is modeled by R = 6e 12.77x, where x is the blood alcohol concentration and R, given as percent, is the risk of having a car accident. What blood alcohol concentration corresponds to a 17% risk of car accident? For 17% risk we let R=17. 8 Page
Exponential Growth and Decay Exponential Growth and Decay Models f(t) = A 0 e kt or A = A 0 e kt If k>0, the function models the amount, or size, of a growing entity. A 0 is the original amount, or size, of the growing entity at time t=0, A is the amount at time t, and k is a constant representing the growth time. If k<0, the function models the amount, or size, of a decaying entity. A 0 is the original amount, or size, of the decaying entity at time t=0, A is the amount at time t, and k is a constant representing the decay rate. Example The below graph shows the US population in millions for five selected years from 1970 to 2007. 350 300 250 200 203.3 US Population, 1970-2007 300.9 281.4 248.7 226.5 150 US Population, 1970-2007 100 50 0 1970 1980 1990 2000 2007 9 Page
1. Find an exponential growth function that models the data for 1970 through 2007 2. By which year will the US population reach 312 million 3. Is this example an exponential growth or decay? 10 Page
Example : Carbom-14 decays exponentially with a half-life of approximately 5715 years. The half-life of a substance is the time required for half of a given sample to disintegrate. Thus after 5715 years a given amount of carbor-14 will have decayed to half the original amount 1. Find an exponential decay model for carbon-14 2. In 1947, earthenware jars containing what are known as the Dead Sea Scrolls were found. Analysis indicated that the scroll wrappings contained 76% of their original carbon-14. Estimate the age of the Dead Sea Scrolls. 11 Page
Logistic Growth Models : The mathematical model for limited c logistic growth is given by f(t) = or A = c, 1+ae bt 1+ae bt where a, b, and c are constants with c>0 and b>0 Note that as time increases to infinity then the expression goes to 0. Example The following models describes the number of people, f(t), who have become ill with influenza t weeks after its initial outbreak in a town with 30,000 inhabitants. f(t) = 30,000 1 + 20e 1.5t 1. How many people became ill with the flu when the epidemic began? (t=0) 2. How many people were ill by the end of the forth week? (t=4) 3. What is the limiting size of f(t), the population that became ill? 12 Page
Newton s Law of Cooling The temperature T, of a heated object at time t is given by T = C + (T 0 C)e kt, where C is the constant temperature of the surrounding medium, T 0 is the initial temperature of the heated object, and k is a negative constant that is associated with the cooling object. Example : A cake removed from the oven has a temperature of 210 F. It is left to cool in a room that has a temperature of 70 F. After 30 minutes, the temperature of the cake is 140 F. 1. Use Newton s Law of Cooling to fins a model for the temperature of the cake, T, after t minutes 13 Page
2. What is the temperature of the cake after 40 minutes? 3. When will the temperature of the cake be 90 F? 14 Page
Choosing a Model for Data If the graph of your data increases/ decreases rapidly then begins to level off a bit then choose a logarithmic model y = a + b ln x a > 0 and b > 0 y = a + b ln x a > 0 and b < 0 If the graph of your data increases/ decreases more and more rapidly then choose an exponential model y = ab x a > 0 and b > 1 y = ab x a > 0 and 0 < b < 1 Example : Choose either Logarithmic or Exponential model Number of Weight Loss Surgeries (thousands) 200 150 100 50 Number of Weight Loss Surgeries inthe US 2006, 180 0 2001 2002 2003 2004 2005 2006 2007 Year Number of Weight Loss Surgeries (thousands) 15 Page
7 World Population 1950-2006 6 Population (billions) 5 4 3 2 1 Population (billions) 0 1940 1950 1960 1970 1980 1990 2000 2010 Year Expressing y = ab x in base e y = ab x is equivalent to y = ae (lnb)x Example Rewrite the following 1. g(x) = 2.569(1.017) x 2. y = 4(7.8) x 3. h(x) = 4.5(0.6) x 16 Page