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Logarithmic Expressions and Equations

Learn to simplify logarithmic expressions using properties, solve equations involving logarithms and exponents, and judge validity of arguments. Practice using Product, Quotient, and Power Properties of logarithms.

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Logarithmic Expressions and Equations

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  1. Logarithmic Expressions and Equations 14.0 Students understand and use the properties of logarithms to simplify logarithmic numeric expressions and to identify their approximate values. 11.1 Students understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. 11.2 Students judge the validity of an argument according to whether the properties of real numbers, exponents, and logarithms a have been applied correctly at each step.

  2. Pre-requisite Check Simplify Answers: 5, 1, 4, 24

  3. Objectives • Use properties of logarithms to evaluate, expand, or condense such expressions to be able to solve problems involving logarithms. • Solve exponential and logarithmic equations. • Rewrite equivalent logarithm expression by changing base (if we have time, else next week)

  4. Objective 1: Logarithm Properties • Let b, m, and n be positive numbers such that b≠1. • Product Property • Quotient Property • Power Property

  5. Example 1 Use log7 2 0.356 and log7 5 0.827 to find the value of the expression to the nearest thousandth. ≈ ≈ 2 2 a. b. c. log7 log7 10 log7 25 5 5 SOLUTION a. log7 – Quotient property log7 2 log7 5 = – 0.356 0.827 Use the given values of log7 2 and log7 5. ≈ – 0.471 = Simplify. Use Properties of Logarithms

  6. Example 1 = ( ) Product property log7 2 log7 5 2 5 + • = b. log7 10 Use the given values of log7 2 and log7 5. 0.356 0.827 + ≈ 1.183 = Simplify. c. log7 25 log7 52 Express 25 as a power. = 2 log7 5 Power property = Use the given value of log7 5. ( ) 0.827 2 ≈ Simplify. 1.654 = Use Properties of Logarithms log7 Express 10 as a product.

  7. Example 2 3x 3x a. b. b. log4 5x2 log7 log7 y y SOLUTION a. log4 5x2 log45 log4x2 Product property + = Power property log452 log4x + = log73x log7y – Quotient property = Expand a Logarithmic Expression Expand the expression. Assume all variables are positive.

  8. Example 2 Expand a Logarithmic Expression log73 log7xlog7y – + Product property =

  9. Example 3 a. b. log 16 2 log 2 3 log 5 log 4 – + SOLUTION a. log 16 2 log 2 log 16 log 22 – – Power property = Quotient property = Simplify. log 4 = 16 log 22 Condense a Logarithmic Expression Condense the expression.

  10. Example 3 b. 3 log 5 log 4 log 53 log 4 + Power property + = Product property log ( ) 53 4 • = Simplify. log 500 = Condense a Logarithmic Expression

  11. Expand and Condense Logarithmic Expressions Checkpoint 3 1. log5 21 log5 7 2. log5 9 ANSWER 1.366 3. log5 49 ANSWER 2.418 4. ANSWER – 0.526 ANSWER 1.892 Use log5 3 0.683 and log5 7 1.209 to find the value of the expression to the nearest thousandth. (Calculator!!) ≈ ≈

  12. Expand and Condense Logarithmic Expressions Checkpoint 5. log2 5x log2 x ANSWER log2 5 + 5x 6. log 2x 3 3 log x ANSWER log 2 + log3 7 7. ANSWER 8. log6 4 2 log6x log6y ANSWER 4x2 – + log6 y log3 5 log3x log3 7 – + Expand the expression. Assume all variables are positive.

  13. Expand and Condense Logarithmic Expressions Checkpoint x3 9. log5 12 log5 4 ANSWER – log5 3 y 10. log2 7 log2 5 ANSWER + log2 35 11. log 4 2log 3 log 36 ANSWER + 12. 3 log x log y log ANSWER – Condense the expression. Assume all variables are positive.

  14. Objective 2: Solving Equations 1. Equal Powers Property 2. Equal Logarithms Property • For b>0 and b≠1, IFF x=y • Example: If , then x=5 • For positive numbers b, x, and y where b≠1, IFF x=y • Example:

  15. 1. Equal Powers Property Solve the equation

  16. Take Common Logarithm of Each Side • Solve • You try, solve

  17. The Answer to

  18. 2. Equal Logarithms Property Solve • Solution:

  19. Exponentiate Both Sides Solve • Solution:

  20. Objective 3: Change Base Formula • Change-of-Base Formula • Let x, b, and c be positive numbers such that b≠1 and c≠1. Then,

  21. Change Base Formula Change-of-base 6 for Answer: Change-of-base to the common logarithm of Answer:

  22. Conclusion Summary Assignment • How do you solve logarithmic equations? • If the equations can be rewritten so they have the same base and if a single variable appears as an exponent, take the logarithm of each side and solve for the variable; if the two sides of the equation can be written as logarithms to the same base, set the logarithms equal. • Pg445 #(22-33, 35-57 ODD) • Pg452 #(27-33, 37-51 ODD) • Mid-Unit Quiz Monday/Tuesday • UNIT Exam next week on Thursday/Friday.

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