Properties of Logarithms Skill 15 Objectives Rewrite logarithms

Properties of Logarithms Skill #15

Objectives… �Rewrite logarithms with different bases. �Use properties of logarithms to evaluate or rewrite logarithmic expressions. �Use properties of logarithms to expand or condense logarithmic expressions.

Change of Base Most calculators have only two types of log keys, one for common logarithms (base 10) and one for natural logarithms (base e). These are the most commonly used. To evaluate logarithms to other bases use the change-of-base formula.

Change of Base

Example–Changing Bases Using Common Logarithms Log 425 2. 23 Log 212 3. 58

Example–Changing Bases Using Common Logarithms Log 632 Log 313

Properties of Logarithms… Product Property Quotient Property Power Property

Example–Using Properties of Logarithms Write each logarithm in terms of ln 2 and ln 3. ln 6 = ln(2 3) = ln 2 + ln 3 ln = ln 2 – ln 27 = ln 2 – ln 33 = ln 2 – 3 ln 3

Rewriting Logarithmic Expressions The properties of logarithms are useful for rewriting logarithmic expressions in forms that simplify the operations of algebra.

Example–Expanding Logarithmic Expressions Use the properties of logarithms to expand each expression. a. log 45 x 3 y b. ln Solution: log 45 x 3 y = log 45 + log 4 x 3 + log 4 y = log 45 + 3 log 4 x + log 4 y

Example–Solution

Rewriting Logarithmic Expressions We can also use properties of logarithms to condense logarithmic expressions.

Example–Condensing Logarithmic Expressions a. log 10 x + 3 log 10(x + 1) = log 10 x 1/2 + log 10(x + 1)3 b. 2 ln(x + 2) – ln x = ln(x + 2)2 – ln x

Example–Condensing Logarithmic Expressions

15 Properties of Logarithms � Summarize � Questions? � Homework ◦ Worksheet Notes

Example 7 – Finding a Mathematical Model The table shows the mean distance x from the sun and the period y (the time it takes a planet to orbit the sun) for each of the six planets that are closest to the sun. In the table, the mean distance is given in astronomical units (where the Earth’s mean distance is defined as 1. 0), and the period is given in years.

Example 7 – Finding a Mathematical Model The points in the table are plotted in Figure 3. 22. Find an equation that relates y and x.

Example 7 – Solution From Figure 3. 22, it is not clear how to find an equation that relates y and x. To solve this problem, take the natural log of each of the x- and y-values in the table. This produces the following results.

Example 7 – Solution Now, by plotting the points in the table, you can see that all six of the points appear to lie in a line, as shown in Figure 3. 23. To find an equation of the line through these points, you can use algebraic method. Choose any two points to determine the slope of the line. Using the two points (0. 421, 0. 632) and (0, 0) you can determine that the slope of the line is

Example 7 – Solution By the point-slope form, the equation of the line is Where Y = ln y and X = ln x. You can therefore conclude that