Graph y = b for 0 < b < 1 x EXAMPLE 1 Graph y = 1 2 x SOLUTION STEP 1 Make a table of values STEP 2 Plot the points from the table. STEP 3 Draw, from right to left, a smooth curve that begins just above the x-axis, passes through the plotted points, and moves up to the left.

EXAMPLE 2 Graph y = ab for 0 < b < 1 x Graph the function. a. Graph y = 2 1 4 x SOLUTION Plot (0, 2) and .Then, from right to left, draw a curve that begins just above the x-axis, passes through the two points, and moves up to the left. 1, 1 2 a.

EXAMPLE 2 Graph y = ab for 0 < b < 1 x Graph the function. b. Graph y = –3 2 5 x SOLUTION Plot (0, –3) and . Then,from right to left, draw a curve that begins just below the x-axis, passes through the two points,and moves down to the left. b. 1, – 6 5

Graph y = ab + k for 0 < b < 1 EXAMPLE 3 x – h EXAMPLE 3 Graph y = 3 –2. State the domain and range. 1 2 x+1 SOLUTION Begin by sketching the graph of y = , which passes through (0, 3) and . Then translate the graph left 1 unit and down 2 units .Notice that the translated graph passes through (– 1, 1) and 3 1 2 x 1, – 1 0,

Graph y = ab + k for 0 < b < 1 EXAMPLE 3 x – h EXAMPLE 3 The graph’s asymptote is the line y = –2. The domain is all real numbers, and the range is y > –2.

EXAMPLE 4 Solve a multi-step problem A new snowmobile costs $4200. The value of the snowmobile decreases by 10% each year. Snowmobiles • Write an exponential decay model giving the snowmobile’s value y (in dollars) after t years. Estimate the value after 3 years. • Graph the model. • Use the graph to estimate when the value of the snowmobile will be $2500.

Solve a multi-step problem EXAMPLE 4 Solve a multi-step problem SOLUTION STEP 1 The initial amount is a = 4200 and the percent decrease is r = 0.10. So, the exponential decay model is: y = a(1 – r) t Write exponential decay model. = 4200(1 – 0.10) t Substitute 4200 for a and 0.10 for r. = 4200(0.90) t Simplify. When t = 3, the snowmobile’s value is y = 4200(0.90)3 = $3061.80.

EXAMPLE 4 Solve a multi-step problem STEP 2 The graph passes through the points (0, 4200) and (1, 3780).It has the t-axis as an asymptote. Plot a few other points. Then draw a smooth curve through the points. STEP 3 Using the graph, you can estimate that the value of the snowmobile will be $2500 after about 5 years.

(5 ) ( ) EXAMPLE 1 Simplify natural base expressions Simplify the expression. a. e2 e5 = e2 + 5 = e7 b. 12 e4 3 e3 = e4 – 3 4 = 4e (5 ) c. e –3x 2 = 52 ( e –3x ) 2 = 25 e –6x = 25 e6x

EXAMPLE 2 Evaluate natural base expressions Use a calculator to evaluate the expression. Expression Keystrokes Display a. e4 [ ] ex 4 54.59815003 b. e –0.09 [ ] ex 0.09 0.9139311853

EXAMPLE 3 Graph natural base functions Graph the function. State the domain and range. a. y = 3e 0.25x SOLUTION Because a = 3 is positive and r = 0.25 is positive, the function is an exponential growth function. Plot the points (0, 3) and (1, 3.85) and draw the curve. The domain is all real numbers, and the range is y > 0.

EXAMPLE 3 Graph natural base functions Graph the function. State the domain and range. b. y = e –0.75(x – 2) + 1 SOLUTION a = 1 is positive and r = –0.75 is negative, so the function is an exponential decay function. Translate the graph of y = right 2 units and up 1 unit. e –0.75x The domain is all real numbers, and the range is y > 1.

EXAMPLE 4 Solve a multi-step problem Biology The length l (in centimeters) of a tiger shark can be modeled by the function e –0.178t l = 337 – 276 where t is the shark’s age (in years). • Graph the model. • Use the graph to estimate the length of a tiger shark that is 3 years old.

EXAMPLE 4 Solve a multi-step problem SOLUTION STEP 1 Graph the model, as shown. STEP 2 Use the trace feature to determine that l 175 when t = 3. The length of a 3year-old tiger shark is about 175 centimeters. ANSWER

Model continuously compounded interest EXAMPLE 5 Model continuously compounded interest Finance You deposit $4000 in an account that pays 6% annual interest compounded continuously. What is the balance after 1 year? SOLUTION Use the formula for continuously compounded interest. A = Pert Write formula. = 4000 e0.06(1) Substitute 4000 for P, 0.06 for r, and 1 for t. 4247.35 Use a calculator. The balance at the end of 1 year is $4247.35. ANSWER

EXAMPLE 1 Rewrite logarithmic equations Logarithmic Form Exponential Form a. = 2 log 8 3 23 = 8 b. 4 log 1 = 40 = 1 = c. 12 log 1 121 = 12 4 = –1 1 = d. 1/4 log –1 4

EXAMPLE 2 Evaluate logarithms Evaluate the logarithm. 4 log a. 64 SOLUTION b log To help you find the value of y, ask yourself what power of b gives you y. 4 to what power gives 64? a. 4 log 43 64, so = 3. 64 b. 5 log 0.2 5 to what power gives 0.2? b. = 5–1 0.2, so –1. 0.2 5 log

EXAMPLE 2 Evaluate logarithms Evaluate the logarithm. c. 1/5 log 125 SOLUTION b log To help you find the value of y, ask yourself what power of b gives you y. c. to what power gives 125? 1 5 = –3 1 5 125, so 1/5 log 125 –3. d. 36 log 6 361/2 6, so 36 log 6 = 1 2 . d. 36 to what power gives 6?

EXAMPLE 3 Evaluate common and natural logarithms Expression Keystrokes Display Check a. log 8 8 0.903089987 100.903 8 b. ln 0.3 .3 –1.203972804 0.3 e –1.204

EXAMPLE 4 Evaluate a logarithmic model Tornadoes The wind speed s (in miles per hour) near the center of a tornado can be modeled by where d is the distance (in miles) that the tornado travels. In 1925, a tornado traveled 220 miles through three states. Estimate the wind speed near the tornado’s center. 93 log d + 65 s =

Evaluate a logarithmic model EXAMPLE 4 Evaluate a logarithmic model SOLUTION 93 log d + 65 s = Write function. = 93 log 220 + 65 Substitute 220 for d. 93(2.342) + 65 Use a calculator. = 282.806 Simplify. The wind speed near the tornado’s center was about 283 miles per hour. ANSWER

( ) EXAMPLE 5 Use inverse properties Simplify the expression. a. 10log4 b. 5 log 25x SOLUTION a. 10log4 = 4 b log x = x b. 5 log 25x = ( 52 ) x 5 log Express 25 as a power with base 5. = 5 log 52x Power of a power property 2x = b log bx = x

Find inverse functions EXAMPLE 6 Find inverse functions Find the inverse of the function. a. y = 6 x b. y = ln (x + 3) SOLUTION a. 6 log From the definition of logarithm, the inverse of y = 6 x is y = x. b. y = ln (x + 3) Write original function. x = ln (y + 3) Switch x and y. = ex (y + 3) Write in exponential form. = ex – 3 y Solve for y. ANSWER The inverse of y = ln (x + 3) is y = ex – 3.

EXAMPLE 7 Graph logarithmic functions Graph the function. a. y = 3 log x SOLUTION Plot several convenient points, such as (1, 0), (3, 1), and (9, 2). The y-axis is a vertical asymptote. From left to right, draw a curve that starts just to the right of the y-axis and moves up through the plotted points, as shown below.

EXAMPLE 7 Graph logarithmic functions Graph the function. b. y = 1/2 log x SOLUTION Plot several convenient points, such as (1, 0), (2, –1), (4, –2), and (8, –3). The y-axis is a vertical asymptote. From left to right, draw a curve that starts just to the right of the y-axis and moves down through the plotted points, as shown below.

EXAMPLE 8 Translate a logarithmic graph Graph . State the domain and range. y = 2 log (x + 3) + 1 SOLUTION STEP 1 Sketch the graph of the parent function y = x, which passes through (1, 0), (2, 1), and (4, 2). 2 log STEP 2 Translate the parent graph left 3 units and up 1 unit. The translated graph passes through (–2, 1), (–1, 2), and (1, 3). The graph’s asymptote is x = –3. The domain is x > –3, and the range is all real numbers.