# Objectives: Assignment: To solve exponential P. 253: 9-20 (Some)

Objectives: 1. To solve exponential equations 2. To solve logarithmic equations • • • • • • Assignment: P. 253: 9-20 (Some) P. 254: 25-46 (Some) P. 254: 47-66 (Some) P. 254: 75-88 (Some) P. 254: 89-102 (Some) Homework Supplement You will be able to solve exponential equations If 5x = 57, then what is the value of x? Property of Equality of Exponential Equations If b is a positive real number not equal to 1, then bx = by iff x = y. – 5x = 57 iff x = 7 – This means that one way to solve exponential equations is to make the bases equal so that the exponents are equal. 1 Solve 3 9 x x 3 Solve 9x = 25 This exponential equation cannot be solved using the handy Property of Equal Exponents. We need logarithms. Sometimes you can’t rewrite your equation so that the bases are the same (or maybe you just don’t want to). In that case, try one of the following: 1. Take the log of both sides – – 2. Rewrite the equation in logarithmic form Both of these are really the same thing You may have to use the change of base formula 9 x 25 log 9 9 x log 9 25 Method 1: Take the log9 of both sides. The base of the exponential becomes the base of the log. x log9 25 Use the Inverse Property to get x on the left. log 25 x log 9 Use the Change of Base formula. x 1.465 9 x 25 ln 9 x ln 25 x ln9 ln 25 ln 25 x ln 9 x 1.465 Method 1e: Take the ln or log of both sides. Don’t worry about the original base. Use the Power Property. Divide. The Change of Base formula happens automatically 9 x 25 log9 25 x log 25 x log 9 x 1.465 Method 2: Write the original equation in logarithmic form. Use the Change of Base formula. Solve −3e2x + 16 = 5 Solve each equation. 1. 8x – 1 = 323x – 2 2. 2x = 5 Solve each equation. 1. 79x = 15 2. 4e−0.3x – 7 = 13 You have $1000 burning a hole in your pocket. Your frugal mom has convinced you to put the money into a savings account that earns 2.25% APR, compounded continuously. How long will it take to double your money? A new car costs $25,000. The value of the car decreases by 15% each year. Write an exponential decay model giving the car’s value y (in dollars) after t years. In how many years with the value of the car be ½ the original value? According to NATO, a movie ticket to see Star Wars (Episode IV: A New Hope) in 1977 was an average price of $2.23. In 1997 when Star Wars was re-released to theaters in anticipation of the upcoming prequel trilogy, ticket prices averaged $4.59. Use the growth model y = Pert to find the annual growth rate from 1977 to 1997. You will be able to solve logarithmic equations If log2 x = log2 25, then what is the value of x? Property of Equality of Logarithmic Equations If b, x, and y are positive real numbers with b ≠ 1, then logb x = logb y iff x = y. – log2 x = log2 25 iff x = 25 Solve log4 (2x + 8) = log4 (6x – 12). Since x = y means bx = by, we can exponentiate both sides of an equation using the same base. – Exponentiate means to raise a quantity to a power – Notice the two sides of the equation become the exponents – Essentially rewriting a logarithmic equation as an exponential equation – Check for extraneous solutions (based on domain) Solve log7 (3x – 2) = 2. Solve log6 3x + log6 (x – 4) = 2. Solve each equation. 1. ln (7x – 4) = ln (2x + 11) 2. log2 (x – 6) = 5 Solve each equation. 1. log 5x + log (x – 1) =2 2. log4 (x + 12) + log4 x = 3 Sometimes we want to solve an equation that looks kind of like a quadratic. Well, just treat it like one. e2 x 7e x 12 0 Kind of like ex 3 ex 4 0 ex 3 0 ex 4 0 ex 3 ex 4 x ln 3 x ln 4 x 1.099 x 1.386 k 2 7 k 12 0 k 3 k 4 0 What if the exponential has different bases? Just pick one to take the log of, the other one will become an exponent. 5 34 x 5 7 2 x x 4 2 log 3 7 4 x 5 2x log 3 3 log 3 7 5 2x x 4 x 5 log 3 7 4 log 3 7 2 4 x 5 2 x log3 7 5 x 4 x 2 x log3 7 5 4 log 49 log 3 x 4 2log3 7 5 x 10.929 What if the logarithms have different bases, one of which is a power of the other? Use a change of base—keep the smaller, get rid of the larger. log3 2 x log9 (13x 3) log 3 2 x log 3 2 x log 3 (13 x 3) log 3 9 log 3 (13 x 3) 2 2 log3 2 x log3 (13x 3) log 3 2 x log 3 (13 x 3) 2 2x 2 (13 x 3) 4 x 2 13x 3 4 x 2 13x 3 0 4x 1 x 3 0 x 1/ 4 x3 Solve each equation. Objectives: 1. To solve exponential equations 2. To solve logarithmic equations • • • • • • Assignment: P. 253: 9-20 (Some) P. 254: 25-46 (Some) P. 254: 47-66 (Some) P. 254: 75-88 (Some) P. 254: 89-102 (Some) Homework Supplement

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

### Related manuals

Download PDF

advertisement