MAT 150 – Class #18

Objectives  Graph and evaluate logarithmic functions  Convert equations to logarithmic and exponential forms  Evaluate and apply common logarithms  Evaluate and apply natural logarithms  Apply logarithmic properties

Example Write each of the following exponential equations in logarithmic form. Solution a. 4 2 = 16 b. c. d.

Example Write each of the following logarithmic equations in exponential form. Solution a. log 2 32 = 5 b. log = –5 c. log 8 1 = 0

Example Graph y = log 5 x. Solution x = 5 y y –2 – /25 1/

Example Explain how the graph of each of the following functions compares with the graph of y = log x, find the domain, and graph each function. a. y = log(x + 3) b. y = 4 + log(x – 2) c.

Example (cont) a. y = log(x + 3) Solution The graph of y = log(x + 3) has the same shape as the graph of y = log x, but it is shifted 3 units to the left. Because the graph has been shifted left 3 units, the domain is (–3, ∞).

Example (cont) b. y = 4 + log(x – 2) Solution The graph of y = 4 + log(x - 2) has the same shape as the graph of y = log x, but it is shifted 2 units to the right and 4 units up. The vertical asymptote is x = 2, The domain is (2, ∞).

Example (cont) c. Solution Multiplication of a function by a constant less than 1 compresses the graph by a factor equal to that constant. The domain is (0, ∞).

Example Projections from 2010 to 2050 indicate that the percent of U.S. adults with diabetes (diagnosed and undiagnosed) can be modeled by p(x) = – ln x, where x is the number of years after a. Graph this function. b. Is the function increasing or decreasing? What does this mean in the context of the application? c. What does this model predict the percent of U.S. adults with diabetes will be in 2022? d. Use the graph to estimate the year in which this model predicts the percent will reach 33%.

Example (cont) Solution a. Graph this function. b. Is the function increasing or decreasing? What does this mean in the context of the application? The function is increasing, which means the percent of U.S. adults with diabetes is predicted to increase from 2010 to y= – ln x

Example (cont) c. What does this model predict the percent of U.S. adults with diabetes will be in 2022? The year 2022 is 22 years after 2000, so the percent in 2022 is estimated to be p(22) = – ln 22 ≈ 23.7 d. Use the graph to estimate the year in which this model predicts the percent will reach 33%. The percent is 33% when x ≈ 48.4, during 2049.

Example

Rewrite each of the following expressions as a single logarithm. a. log 5 x + 3log 5 y b. Solution a. log 5 x + 3log 5 y b. = log 5 x + log 5 y 3 = log 5 xy 3

Example (cont) Rewrite each of the following expressions as a single logarithm. c. Solution c.

Assignment Pg #1-11 odd #15-17 odd (Remember to be detailed on graphs.) #31-37 odd #39, 41