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Understanding Exponents and Logarithms: Laws and Applications in Algebra

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This lesson focuses on the foundational concepts of exponents and logarithms, essential for solving complex algebraic problems. Students will explore various exponent laws, such as product and quotient rules, and how these principles translate into logarithmic form. The agenda includes understanding logarithmic definitions, common mistakes, and problem-solving strategies using exponential and logarithmic equations. With an upcoming test on Monday, this comprehensive session aims to prepare students effectively through practice problems and clear explanations of the connections between roots, exponents, and logs.

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Understanding Exponents and Logarithms: Laws and Applications in Algebra

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  1. Calculus-9/21/2011 Take Out: Do Now Sheet, Pencil, Homework DO NOW: Exponent laws handout Do these on answers out on your green sheet under Wednesday (because guess what it is Wednesday) • Agenda • Do Now • Laws of Logarithms • Practice • HW: Log Properties Handout • UPCOMING- Test Moved to MONDAY instead of Friday!! Objectives: Solve complex algebraic problems using laws of logs and exponents. Use the definition of log and exponent to switch between log and exponent form in an algebraic equation.

  2. Exponents, Logs Objectives: Solve complex algebraic problems using laws of logs and exponents. Use the definition of log and exponent to switch between log and exponent form in an algebraic equation.

  3. Exponents • Exponents are repeated multiplication: n times • Example: Objectives: Solve complex algebraic problems using laws of logs and exponents. Use the definition of log and exponent to switch between log and exponent form in an algebraic equation.

  4. Rules for Exponents Rule Example Objectives: Solve complex algebraic problems using laws of logs and exponents. Use the definition of log and exponent to switch between log and exponent form in an algebraic equation.

  5. Practice Objectives: Solve complex algebraic problems using laws of logs and exponents. Use the definition of log and exponent to switch between log and exponent form in an algebraic equation.

  6. More Exponent Rules Objectives: Solve complex algebraic problems using laws of logs and exponents. Use the definition of log and exponent to switch between log and exponent form in an algebraic equation.

  7. Practice Objectives: Solve complex algebraic problems using laws of logs and exponents. Use the definition of log and exponent to switch between log and exponent form in an algebraic equation.

  8. Common Mistakes Objectives: Solve complex algebraic problems using laws of logs and exponents. Use the definition of log and exponent to switch between log and exponent form in an algebraic equation.

  9. ROOTS • Roots don’t count as a separate category, because they are just like exponents. We’ll see why in a second. Objectives: Solve complex algebraic problems using laws of logs and exponents. Use the definition of log and exponent to switch between log and exponent form in an algebraic equation.

  10. Root – Exponent Connection Objectives: Solve complex algebraic problems using laws of logs and exponents. Use the definition of log and exponent to switch between log and exponent form in an algebraic equation.

  11. Practice Objectives: Solve complex algebraic problems using laws of logs and exponents. Use the definition of log and exponent to switch between log and exponent form in an algebraic equation.

  12. Logs • I want you to be able to use logs to solve for a variable. Things to Remember… If you have an exponential equation with a # base use logs to solve. If you have an exponential equation with base e use natural log (ln) to solve. Objectives: Solve complex algebraic problems using laws of logs and exponents. Use the definition of log and exponent to switch between log and exponent form in an algebraic equation.

  13. Logs • Basic Definition of a log: Objectives: Solve complex algebraic problems using laws of logs and exponents. Use the definition of log and exponent to switch between log and exponent form in an algebraic equation.

  14. More Log Rules-Inverse Properties BASE a: BASE e: Objectives: Solve complex algebraic problems using laws of logs and exponents. Use the definition of log and exponent to switch between log and exponent form in an algebraic equation.

  15. How can we use this in an algebraic context? • Whenever the variable you are looking for is in the exponent, we need to use logs Objectives: Solve complex algebraic problems using laws of logs and exponents. Use the definition of log and exponent to switch between log and exponent form in an algebraic equation.

  16. Example 2-using inverse property Objectives: Solve complex algebraic problems using laws of logs and exponents. Use the definition of log and exponent to switch between log and exponent form in an algebraic equation.

  17. Example 3- Using Change of Base Rule Objectives: Solve complex algebraic problems using laws of logs and exponents. Use the definition of log and exponent to switch between log and exponent form in an algebraic equation.

  18. Practice-Now you try either use change of base or inverse property to solve for x • e2x = 10 • 54x + 1 = 15 • 5 ex + 1 = 30 • ex/5 + 4 = 7 • 32x = 40 Objectives: Solve complex algebraic problems using laws of logs and exponents. Use the definition of log and exponent to switch between log and exponent form in an algebraic equation.

  19. Rules of Logarithms • Since a logarithm is simply an exponent which is just being written down on the line, we expect the logarithm laws to work the same as the rules for exponents, and luckily, they do ExponentsLogarithms bm × bn = bm+nlogbxy = logbx + logby bm ÷ bn = bm-nlogb (x/y) = logbx − logby (bm)n = bmnlogb (xn) = nlogbx b1 = blogb (b) = 1 b0 = 1 logb (1) = 0 Objectives: Solve complex algebraic problems using laws of logs and exponents. Use the definition of log and exponent to switch between log and exponent form in an algebraic equation.

  20. Example 1 Apply product property Change into exponential form to solve Simplify Reduce 1 side to zero to solve the quadratic Factor Solutions!! Objectives: Solve complex algebraic problems using laws of logs and exponents. Use the definition of log and exponent to switch between log and exponent form in an algebraic equation.

  21. Example 2 Objectives: Solve complex algebraic problems using laws of logs and exponents. Use the definition of log and exponent to switch between log and exponent form in an algebraic equation.

  22. Example 3 Product Property of Logs Switch into exponential form Simplify Get rid of the fraction by multiplying (x-4) Solve for x Objectives: Solve complex algebraic problems using laws of logs and exponents. Use the definition of log and exponent to switch between log and exponent form in an algebraic equation.

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