Chapter 4, Continued. 4.3 Laws of Logarithms. 1. log a (AB) = log a A + log a B. 2. log a ( A B ) = log a A log a B. 3. log a (A c ) = C log a A

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1 Chapter 4, Continued 4.3 Laws of Logarithms 1. log a (AB) = log a A + log a B 2. log a ( A B ) = log a A log a B 3. log a (A c ) = C log a A : Evaluate the following expressions. log log log log 2 5 log Use Laws of Logarithms to expand the following expressions. ( ) x 2 log a yz 3 ln 3x 2 (x + 1) 10 ln 3 3r 2 s ln x

2 Use Laws of Logarithms to combine the following expressions. (Write as a single logarithm.) 1 2 log 5(x 2 1) log 5 (x 1) 2(log 3 x log 3 (y + 1) 3 log 3 z) The following are common errors that students make with logarithms and are NOT true. log a (A + B) = log a A + log a B log a (AB) = (log a A)(log a B) log a ( A B ) = log a A log a B (log a A) c = c log a A log a A log a B = A B Sometimes it is necessary to change the base of a logarithm so that you can evaluate it on a calculator or so that you can write it in terms of the common or natural logarithm. Change of Base Formula: log b x = log a x log a b. Write log 5 2 with common logarithms and evaluate to 4 decimal places. Write log 6 92 with natural logarithms and evaluate to 4 decimal places. 2

3 4.4 Exponential and Logarithmic Equations To solve exponential equations, 1. Isolate the exponential expression on one side. 2. Take the logarithm of both sides and simplify using Laws of Logarithms. (If the exponential expression has base e, use ln.) 3. Solve for the variable. : 2 x = 8 2x+1 2e 12x+4 = x 1 = 5 e 2x + 3e x 4 = 0 5 x+1 = 2 3x e 3x 6e 2x + 9e x = 0 3

4 To solve logarithmic equations, 1. Isolate the logarithmic term on one side. You may have to use logarithm laws to do this. 2. Write the equation in exponential form. OR Exponentiate both sides. 3. Solve for the variable 4. Check for extraneous solutions. ln(2 + x) = 4 log 3 (2 x) 3 = 0 log 2 x + log 2 (x 7) = 3 2 log x = log 2 + log(3x 4) 4

5 Applications: If $6500 is invested in an account at 8% per year compounded semiannually. How long will it take for the amount in the account to be $7500? A man invests $6500 in an account that pays 6% interest per year, compounded continuously. How long will it take for his investment to be $8000? Suppose the temperature T of an engine after t minutes is given by the equation ( ) T 20 ln = 0.11t 200 Solve this equation for T. Ebbinghaus Law of Forgetting states that if a task is learned at a performance level P 0, then after a time interval t the performance level P satisfies log P = log P 0 c log(t + 1) where c is a constant that depends on the type of task and t is measured in months. Solve for P. 5

6 4.5 Modeling with Exponential and Logarithmic Functions A population that grows exponentially increases according to the model n(t) = n 0 e rt where n(t)=population at time t, n 0 =initial size of the population, r=relative rate of growth as a decimal, and t=time. The number of a certain species of fish is modeled by the function n(t) = 12e 0.012t where t is measured in years and n(t) is measured in millions. What is the initial number of fish? What is the relative rate of growth of the fish population (as a percentage)? What will the fish population be after 5 years? After how many years will the number of fish reach 30 million? A culture starts with 1500 bacteria. After 45 minutes the count is doubled. Find a function that models the number of bacteria n(t) after t hours. 6

7 Exponential functions can also model exponential decay, specifically radioactive decay. The half-life of a radioactive substance is the time it takes for half the mass to decay. If m 0 is the initial mass of a radioactive substance with half-life h, then the mass remaining at time t is modeled by the function m(t) = m 0 e rt Solve for r in terms of the half-life h. The half-life of radium-226 is 1600 years. Suppose we have a 22-mg sample. Find a function that models the mass remaining after t years. After how long will only 18 mg of the sample remain? After 3 days a sample of radon-222 has decayed to 58% of its original amount. What is the half-life of radon-222? Some applications of logarithmic functions include the ph scale, the magnitude of earthquakes, and the decibel scale. You can read more about these in your textbook on pp

Algebra 2 (Standard) DIA #6

Name: Class: Date: Algebra 2 (Standard) DIA #6 Multiple Choice Identify the choice that best completes the statement or answers the question.. An initial population of 865 quail increases at an annual