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Mastering Logarithmic Expressions: Techniques and Examples

Explore properties of logarithms, expand and combine expressions, use change of base formula, and learn to simplify logarithmic terms with real numbers. Practice with various examples and solutions.

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Mastering Logarithmic Expressions: Techniques and Examples

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  1. Chapter 5 Exponential and Logarithmic Functions

  2. Properties of Logarithms 5.5 Apply basic properties of logarithms Expand and combine logarithmic expressions Use the change of base formula

  3. Properties of the Logarithm For positive numbers m, n, and a ≠ 1 and any real number r :

  4. Expand each expression. Write your answers without exponents. Solution Example: Expanding logarithmic expressions

  5. Example: Expanding logarithmic expressions

  6. Example: Expanding logarithmic expressions

  7. Write each expression as the logarithm of a single expression. Example: Combining terms in logarithmic expressions

  8. Example: Combining terms in logarithmic expressions

  9. Write each expression as the logarithm of a single expression. Example: Combining terms in logarithmic expressions

  10. Example: Combining terms in logarithmic expressions

  11. Example: Combining terms in logarithmic expressions

  12. Example: Combining terms in logarithmic expressions

  13. Change of Base Formula Let x, a ≠ 1, and b ≠ 1 be positive real numbers. Then

  14. Use a calculator to approximate each expression to the nearest thousandth. Solution Example: Applying the change of base formula Slide 5.5 - 14

  15. Example: Applying the change of base formula

  16. Solve the equation graphically. Solution Graph Y1 = log(X^3 + X – 1)/log(2) and Y2 = 5. Their graphs intersect near the point (3.104, 5). Solution is x ≈ 3.104. Example: Using the change of base formula for graphing

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