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5/24/2017 1:55 AM 7.5 - Exp and Log Equations and Inequalities 1 Product: Quotient: Power: a a a n m n a a m a 5/24/2017 1:55 AM n–m (a ) a n m n+m n* m 7.5 - Exp and Log Equations and Inequalities 2 Product: logb ( x) logb ( y) logb ( xy) x Quotient: log b x log b y log b y Power: 5/24/2017 1:55 AM log b (a ) p log b ( a) p 7.5 - Exp and Log Equations and Inequalities 3 If each equation on both sides are exponents: 1. Rewrite both sides by “log”-ing it 2. Use exponent and/or logarithmic rules 3. Solve algebraically, Round to 4 decimal places 4. Check 5/24/2017 1:55 AM 7.5 - Exp and Log Equations and Inequalities 4 Solve and Check: 98 – x = 27x – 3 log 9 8 – x = log 27 x – 3 Rewrite both sides by “log”-ing it (8 – x) log 9 = (x – 3) log 27 log 9 log 27 (8 x) ( x 3) log 9 log 9 8 – x = (x – 3) (1.5) 8 – x = (1.5)x – 3 8 – x = 1.5x – 4.5 x=5 5/24/2017 1:55 AM Follow the log rules; Power Rules Use Algebra to solve Distribute 1.5 to x - 3 Solve for x. Answer. 7.5 - Exp and Log Equations and Inequalities 5 Solve and Check: 98 – x = 27x – 3 9 8 – x = 27 x – 3 2(8 x ) 3 3( x 3) 3 16 – 2x = 3x –9 x=5 5/24/2017 1:55 AM Since 27 is a base of 3, apply it to both sides Use Algebra to solve Solve for x Answer. 7.5 - Exp and Log Equations and Inequalities 6 Solve and Check: 8 x = 2 x + 6 x=3 5/24/2017 1:55 AM 7.5 - Exp and Log Equations and Inequalities 7 Solve and Check: 43x–1 = 8x+1 x = 5/3 5/24/2017 1:55 AM 7.5 - Exp and Log Equations and Inequalities 8 Solve and Check: 4x – 1 = 5 log 4 x – 1 = log 5 Rewrite both sides by “log”-ing it (x – 1) log 4 = log 5 Follow the log rules; Power Rules log 4 log 5 ( x 1) log 4 log 4 Use Algebra to solve x – 1 ≈ 1.1610 x ≈ 2.1610 5/24/2017 1:55 AM Solve for x; Round to four decimal places Answer. 7.5 - Exp and Log Equations and Inequalities 9 Solve and Check: 32x–1 = 20 x ≈ 1.8634 5/24/2017 1:55 AM 7.5 - Exp and Log Equations and Inequalities 10 If each equation on one side shows a log.: 1a. Rewrite the equation in exponential form 1b. Use exponent and/or logarithmic rules (including Change of Base) 2. Solve algebraically, Round to 4 decimal places 3. Check 5/24/2017 1:55 AM 7.5 - Exp and Log Equations and Inequalities 11 Solve : log7(5x + 3) = 3 3 7 = 5x + 3 343 = 5x + 3 5x = 340 x = 68 5/24/2017 1:55 AM Rewriting the equation in exponential form Use Algebra to solve for x Solve for x. Answer. 7.5 - Exp and Log Equations and Inequalities 12 Solve : log6(2x – 1) = –1 x = 7/12 5/24/2017 1:55 AM 7.5 - Exp and Log Equations and Inequalities 13 Solve : log4100 – log4(x + 1) = 1 Can this equation be written in Exponential Form? 100 log 4 1 x 1 100 1 4 x 1 100 4 x 1 4( x 1) 100 x = 24 5/24/2017 1:55 AM NO Write problem using Log properties Rewrite equation using exponential form to solve Solve for x.; cross multiply Answer. 7.5 - Exp and Log Equations and Inequalities 14 Solve : log12x + log12(x + 1) = 1 Why can’t x = –4? ---------------------Plug –4 into original equation. ---------------------- x=3 5/24/2017 1:55 AM 7.5 - Exp and Log Equations and Inequalities The answer is undefined. 15 Solve for x: 1. 23x = 15 2. 7–x = 21 3. log5x 4 = 8 4. 3 = log 8 + 3log x 5/24/2017 1:55 AM 7.5 - Exp and Log Equations and Inequalities 16 Suppose a bacteria culture doubles in size every hour. How many hours will it take for the number of bacteria to exceed 1,000,000? At hour 0, there is one bacterium, or 20 bacteria. At hour one, there are two bacteria, or 21 bacteria, and so on. So, at hour n there will be 2n bacteria. Solve 2n > 106 Write 1,000,000 in scientific annotation. log 2n > log 106 Take the log of both sides. 5/24/2017 1:55 AM 7.5 - Exp and Log Equations and Inequalities 17 Suppose a bacteria culture doubles in size every hour. How many hours will it take for the number of bacteria to exceed 1,000,000? nlog 2 > log 106 Use the Power of Logarithms. nlog 2 > 6 log 106 is 6. 6 log 2 6 n> 0.301 Divide both sides by log 2. n > ≈ 19.94 Round up to the next whole number. n> Evaluate by using a calculator. It will take about 20 hours for the number of bacteria to exceed 1,000,000. 5/24/2017 1:55 AM 7.5 - Exp and Log Equations and Inequalities 18 Worksheet Pg 526 8, 21 – 33 all NOT 27 Non-Calc Quiz Friday 5/24/2017 1:55 AM 7.5 - Exp and Log Equations and Inequalities 19