College Algebra Chapter 4 Exponential and Logarithmic Functions Section 4.3 Logarithmic Functions

1. Convert Between Logarithmic and Exponential Forms 2. Evaluate Logarithmic Expressions 3. Apply Basic Properties of Logarithms 4. Graph Logarithmic Functions 5. Use Logarithmic Functions in Applications

Convert Between Logarithmic and Exponential Forms Logarithmic Function: If x and b are positive real numbers such that b ≠ 1, then is called the logarithmic function with base b where

Examples 1 – 4: Write each equation in exponential form. 1. 2. 3. 4.

Examples 5 – 7: Write each equation in logarithmic form. 5. 6. 7.

1. Convert Between Logarithmic and Exponential Forms 2. Evaluate Logarithmic Expressions 3. Apply Basic Properties of Logarithms 4. Graph Logarithmic Functions 5. Use Logarithmic Functions in Applications

Examples 8 – 10: Evaluate each logarithmic expression. 8. 9. 10.

Evaluate Logarithmic Expressions Common logarithmic function: Natural logarithmic function:

Examples 11 – 14: Evaluate each expression. 11. 12. 13. 14.

Examples 15 – 18: Use your calculator to find the approximate value. Round the answer to 4 decimal places. Check your answer by using the exponential form. Example: 15. 16. 17. 18.

1. Convert Between Logarithmic and Exponential Forms 2. Evaluate Logarithmic Expressions 3. Apply Basic Properties of Logarithms 4. Graph Logarithmic Functions 5. Use Logarithmic Functions in Applications

Apply Basic Properties of Logarithms

Examples 19 – 27: Simplify each expression. 19. 20. 21. 22. 23. 24. 25. 26. 27.

1. Convert Between Logarithmic and Exponential Forms 2. Evaluate Logarithmic Expressions 3. Apply Basic Properties of Logarithms 4. Graph Logarithmic Functions 5. Use Logarithmic Functions in Applications

Example 28: Graph (rewrite into exponential form and select values for y first) exponential form:

Example 29: Graph If h > 0, shift to the right. If h < 0, shift to the left. Graph If a < 0 reflect across the x-axis. Shrink vertically if 0 < |a| < 1. Stretch vertically if |a| > 1. If k > 0, shift upward. If k < 0, shift downward.

Graph Logarithmic Functions

Example 30: Give the domain and range in interval notation. Determine the vertical asymptote.

Example 31: Give the domain and range in interval notation. Determine the vertical asymptote.

Example 32: Give the domain and range in interval notation. Determine the vertical asymptote.

1. Convert Between Logarithmic and Exponential Forms 2. Evaluate Logarithmic Expressions 3. Apply Basic Properties of Logarithms 4. Graph Logarithmic Functions 5. Use Logarithmic Functions in Applications

Example 32: The absolute magnitude, M, of a star is the apparent magnitude, m, a star would have if it were placed 10 parsecs from earth. The lower the value of the magnitude, the brighter the star. Our sun has an apparent magnitude of –26.74. The brightest star in our night sky is Sirius, the Dog Star, with an apparent magnitude of –1.44. The sun appears so bright because it is very close (astronomically speaking). The formula relates a star’s absolute magnitude, apparent magnitude, and its distance, d, from earth in parsecs.

Example 32 continued: If Sirius is 2.637 parsecs from earth and the sun is parsecs from earth, what is the absolute magnitude of each star?