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Dive deep into solving exponential & logarithmic equations with step-by-step examples, tips, and practice problems. Learn to tackle both exponential and logarithmic equations using algebraic techniques, logarithmic rules, and exponential properties. Enhance your problem-solving skills and ensure accuracy with detailed checks. This comprehensive guide covers various types of equations, from simple to complex, to help you excel in your math studies.
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Exponential and Logarithmic Equations 7.5 - Exp and Log Equations and Inequalities
Review - Exponents • Product: • Quotient: • Power: n + m n – m n* m 7.5 - Exp and Log Equations and Inequalities
Review - Logs • Product: • Quotient: • Power: 7.5 - Exp and Log Equations and Inequalities
Solving Exponential Equations If each equation on both sides are exponents: • Rewrite both sides by “log”-ing it • Use exponent and/or logarithmic rules • Solve algebraically, Round to 4 decimal places • Check 7.5 - Exp and Log Equations and Inequalities
Example 1 Solve and Check: 98 – x=27x – 3 log 9 8 – x=log 27 x –3 Rewrite both sides by “log”-ing it Follow the log rules; Power Rules (8 – x) log 9 =(x – 3) log 27 Use Algebra to solve 8 – x = (x – 3) (1.5) Distribute 1.5 to x -3 8 – x = (1.5)x – 3 Solve for x. 8 – x = 1.5x – 4.5 x = 5 Answer. 7.5 - Exp and Log Equations and Inequalities
Example 1 (another way) Solve and Check: 98 – x=27x – 3 9 8 – x=27 x –3 Since 27 is a base of 3, apply it to both sides Use Algebra to solve 16 – 2x = 3x –9 Solve for x x = 5 Answer. 7.5 - Exp and Log Equations and Inequalities
Example 2 Solve and Check: 8 x=2 x +6 x = 3 7.5 - Exp and Log Equations and Inequalities
Your Turn Solve and Check: 43x–1 = 8x+1 x = 5/3 7.5 - Exp and Log Equations and Inequalities
Example 3 Solve and Check: 4x– 1=5 log 4 x– 1=log 5 Rewrite both sides by “log”-ing it Follow the log rules; Power Rules (x – 1) log 4 =log 5 Use Algebra to solve x – 1 ≈ 1.1610 Solve for x; Round to four decimal places x ≈ 2.1610 Answer. 7.5 - Exp and Log Equations and Inequalities
Your Turn Solve and Check: 32x–1 = 20 x ≈ 1.8634 7.5 - Exp and Log Equations and Inequalities
Solving Logarithmic Equations 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 7.5 - Exp and Log Equations and Inequalities
Example 6 Solve : log7(5x + 3) = 3 Rewriting the equation in exponential form 73 =5x + 3 343 = 5x + 3 Use Algebra to solve for x 5x = 340 Solve for x. x = 68 Answer. 7.5 - Exp and Log Equations and Inequalities
Your Turn Solve : log6(2x – 1) = –1 x = 7/12 7.5 - Exp and Log Equations and Inequalities
Example 7 Solve : log4100– log4(x + 1) = 1 NO Can this equation be written in Exponential Form? Write problem using Log properties Rewrite equation using exponential form to solve Solve for x.; cross multiply x = 24 Answer. 7.5 - Exp and Log Equations and Inequalities
Example 8 Solve : log12x + log12(x + 1) = 1 Why can’t x = –4? ---------------------- Plug –4 into original equation. ---------------------- The answer is undefined. x = 3 7.5 - Exp and Log Equations and Inequalities
Your Turn Solve for x: 1. 23x = 15 2. 7–x = 21 3. log5x 4 = 8 4. 3 = log 8 + 3log x 7.5 - Exp and Log Equations and Inequalities
Example 9 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. 7.5 - Exp and Log Equations and Inequalities
6 6 log 2 0.301 n > n > Example 9 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. Divide both sides by log 2. Evaluate by using a calculator. n > ≈ 19.94 Round up to the next whole number. It will take about 20 hours for the number of bacteria to exceed 1,000,000. 7.5 - Exp and Log Equations and Inequalities
Assignment Worksheet Pg 526 8, 21 – 33 all NOT 27 Non-Calc Quiz Friday 7.5 - Exp and Log Equations and Inequalities