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

Concepts 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

Definition of a 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

Examples 5 – 7: Write each equation in logarithmic form

Concepts 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

Common and Natural Logarithmic Functions Common logarithmic function: Natural logarithmic function:

Examples 11 – 14: Evaluate each expression

Example: 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

Concepts 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

Basic Properties of Logarithms

21. Examples 19 – 27: Simplify each expression

Concepts 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 the logarithmic function. (rewrite into exponential form and select values for y first) exponential form:

Graphs of Exponential and Logarithmic Functions

Example 29: Use transformations to graph the function. 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. If h > 0, shift to the right. If h < 0, shift to the left.

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.

Concepts 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 – 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 parsecs from earth and the sun is parsecs from earth, what is the absolute magnitude of each star?