MHF4U Unit 7 Solving Log and Exp Functions.notebook December 11, 2014 MHF4U Unit #7: Solving Exponential and Logarithmic Functions SECTION Page Numbers Questions Prerequisite Skills p. 368 - 369 # 1ab, 2be, 3bc, 4, 5abcdef, 6aceg, 7, 8abch, 9abcd, 10abdh, 11 7.1 p. 368 - 369 # 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, *15a p. 375 - 377 # 2 11, 13, 15 p. 405 p. 384 - 386 #7 # 1ac, 2ac, 3bd, 4, 5, 6, 7, 9, 10, 11 p. 391 - 392 p. 408 - 409 # 1bdf, 2bdef, 3acde, 5, 6, 7a, 8, 9, 11a p. 410 # 1 6, 9 7.2 7.3 7.4 # 1, 2, 3, 4, 5, 6a, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17 Review Note: Questions with an asterix * are extra questions that are optional for the daily homework. However, they are potential “extended-type” questions that could be included on a unit test. Lesson 7.1 Equivalent Forms of Exponential Equations Learning Goal: Today I will solve exponential equations in one variable by determining a common base and by using logarithms, recognizing that logarithms base 10 are commonly used. Catholic Graduate Expectation: Today I will think reflectively and creatively to evaluate situations and solve problems. MHF4U Unit 7 Solving Log and Exp Functions.notebook December 11, 2014 Exponential functions can be represented in many different ways. It is often useful to express an exponential expression using a different base than the one that is given. Examples: 1. Write each expression as a single power of the base indicated. a) 643, base 4 e) b) 643, base 2 , base 3 f) , base 3 c) , base 2 g) 3, base 7 d) h) 53, base 4 Solving Exponential Equations by Changing the Base Express each side of the equation with the same base. Set the exponents equal to each other and solve. Examples: 2. Solve each equation. a) b) , base 3 c) MHF4U Unit 7 Solving Log and Exp Functions.notebook December 11, 2014 3. Consider the equation . a) Solve this equation by expressing both sides as powers of a common base. b) Solve the same equation by taking the logarithm of each side. 4. Solve 5. a) Solve each equation. Give exact answers only. i) 5 = 10x ii) 3 = 10x iii) 6 = 10x b) Use your answers in part a) to state a formula that could be used to solve the equation b = 10x for x, given any base, b. MHF4U Unit 7 Solving Log and Exp Functions.notebook December 11, 2014 Lesson 7.2 Techniques for Solving Exponential Equations Learning Goal: Today I will solve problems involving exponential equations algebraically, including problems arising from real-world applications. Catholic Graduate Expectation: Today I will think reflectively and creatively to evaluate situations and solve problems. An equation maintains balance when the common logarithm is applied to both sides. The power law of logarithms is a useful tool for solving for a variable that appears as part of an exponent. When a quadratic equation is obtained, methods such as factoring and applying the quadratic formula may be useful. Some algebraic methods of solving exponential equations lead to extraneous roots, which are not valid solutions to the original equation. Examples: 1. Solve the following exponential equations. a) 42x – 1 = 3x + 2 b) 72x – 3 = 2-3x MHF4U Unit 7 Solving Log and Exp Functions.notebook December 11, 2014 Recall: The model for Exponential Growth or Decay is where, = amount as a function of time (final amount) = intial amount (@ time 0) = the ratio/multiplier of growth or decay = time (must be the same unit of time as the period) = period of time for the growth or decay to occur The half-life of a radioactive substance is the time required for a sample of the material to decay to half of its initial amount. Examples: 2. Polonium-218 is a radioactive substance that spontaneously decays into lead-214. One minute after a 100mg sample of polonium-218 is placed into a nuclear chamber, only 80mg remains. Determine the half-life of polonium-218. 3. The compound interest formula modeling the future amount, A, of an investment with initial principal P is A = P(1 + i)n, where i is the interest rate per compounding period, in decimal form, and n is the number of compounding periods. A $3000 investment earns 7.8% interest, compounded monthly. a) Write an equation for the value of the investment as a function of time, in years. b) Determine the value of the investment after 5 years. c) How long will it take for the investment to double in value, to the nearest tenth of a year?
