Properties of Logarithms Section 3.3 Properties of Logarithms

Objective By following instructions students will be able to: Rewrite logarithmic functions with different bases. Use properties of logarithms to evaluate or rewrite logarithmic expressions. Use properties of logarithms to expand or condense logarithmic expressions. Use logarithmic functions to model and solve real-life problems.

Change-of-Base Formula Let a, b, and x be positive real numbers such that and . Then can be converted to a different base using any of the following formulas. Base b Base 10 Base e

Example 1: Change bases using common logarithms. a) b)

Example 2: Change bases using natural logarithms. a) b)

U-TRY #1 Rewrite the logarithm as a multiple of (a) a common logarithm and (b) a natural logarithm. 1) 2) 3) 4)

Properties of Logarithms Let a be positive real numbers such that , and let n be a real number. If u and v are positive real numbers, the following properties are true. 1) 2) 3)

Example 3: Write the logarithms in terms of ln2 and ln3. a) b)

Example 4: Write the logarithms to verify that .

Example 5: Use the properties of logarithms to expand .

Example 6: Use the properties of logarithms to expand .

U-TRY #2 Use the properties of logarithms to write the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) 1) 2) 3) 4)

Example 7: Use the properties of logarithms to condense each logarithmic expression. a) b)

U-TRY #3 Write the expression as the logarithm of a single quantity. 1) 2) 3) 4)

Example 8: The table shows the mean distance from the sun x and the orbital period y of the six planets that are closer to the sun. In the table, the mean distance is given in terms of astronomical units (where the earth’s mean distance is defined as 1.0), and the period is given in terms of years. Find an equation that expresses y as a function of x. Planet Mercury Venus Earth Mars Jupiter Saturn Mean Distance , x 0.387 0.723 1.0 1.524 5.203 9.539 Period, y 0.241 0.615 1.881 11.862 29.458

Revisit Objective Did we… Rewrite logarithmic functions with different bases? Use properties of logarithms to evaluate or rewrite logarithmic expressions? Use properties of logarithms to expand or condense logarithmic expressions? Use logarithmic functions to model and solve real-life problems?

Homework Pg 244#s 1-93 EOO