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Five-Minute Check (over Lesson 7–2) CCSS Then/Now New Vocabulary Key Concept: Logarithm with Base b Example 1: Logarithmic to Exponential Form Example 2: Exponential to Logarithmic Form Example 3: Evaluate Logarithmic Expressions Key Concept: Parent Function of Logarithmic Functions Example 4: Graph Logarithmic Functions Key Concept: Transformations of Logarithmic Functions Example 5: Graph Logarithmic Functions Example 6: Real-World Example: Find Inverses of Exponential Functions Lesson Menu

Solve 42x = 163x – 1. A. x = –1 B. x = C. x = 1 D. x = 2 __ 1 2 5-Minute Check 1

Solve 8x – 1 = 2x + 9. A. x = 10 B. x = 8 C. x = 6 D. x = 4 5-Minute Check 2

Solve 52x – 7< 125. A. x < 6 B. x < 5 C. x < 4 D. x > 3 5-Minute Check 3

Solve A. x ≥ 2 B. x ≥ 1 C. x > 0 D. x ≥ –2 5-Minute Check 4

A money market account pays 5. 3% interest compounded quarterly A money market account pays 5.3% interest compounded quarterly. What will be the balance in the account after 5 years if $12,000 is invested? A. $18,360.00 B. $15,613.98 C. $15,180.00 D. $14,544.00 5-Minute Check 5

Charlie borrowed $125,000 for his small business at a rate of 3 Charlie borrowed $125,000 for his small business at a rate of 3.9% compounded annually for 30 years. At the end of the loan, how much will he have actually paid for the loan? A. $146,250 B. $271,250 C. $389,625.25 D. $393,891.35 5-Minute Check 6

Mathematical Practices 6 Attend to precision. Content Standards F.IF.7.e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f (kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Mathematical Practices 6 Attend to precision. CCSS

You found the inverse of a function. Evaluate logarithmic expressions. Graph logarithmic functions. Then/Now

logarithm logarithmic function Vocabulary

Concept

A. Write log3 9 = 2 in exponential form. Logarithmic to Exponential Form A. Write log3 9 = 2 in exponential form. log3 9 = 2 → 9 = 32 Answer: 9 = 32 Example 1

B. Write in exponential form. Logarithmic to Exponential Form B. Write in exponential form. Answer: Example 1

A. What is log2 8 = 3 written in exponential form? B. 23 = 8 C. 32 = 8 D. 28 = 3 Example 1

B. What is –2 written in exponential form? C. D. Example 1

A. Write 53 = 125 in logarithmic form. Exponential to Logarithmic Form A. Write 53 = 125 in logarithmic form. 53 = 125 → log5 125 = 3 Answer: log5 125 = 3 Example 2

B. Write in logarithmic form. Exponential to Logarithmic Form B. Write in logarithmic form. Answer: Example 2

A. What is 34 = 81 written in logarithmic form? A. log3 81 = 4 B. log4 81 = 3 C. log81 3 = 4 D. log3 4 = 81 Example 2

B. What is written in logarithmic form? D. Example 2

log3 243 = y Let the logarithm equal y. Evaluate Logarithmic Expressions Evaluate log3 243. log3 243 = y Let the logarithm equal y. 243 = 3y Definition of logarithm 35 = 3y 243 = 35 5 = y Property of Equality for Exponential Functions Answer: So, log3 243 = 5. Example 3

Evaluate log10 1000. A. B. 3 C. 30 D. 10,000 Example 3

Concept

A. Graph the function f(x) = log3 x. Graph Logarithmic Functions A. Graph the function f(x) = log3 x. Step 1 Identify the base. b = 3 Step 2 Determine points on the graph. Because 3 > 1, use the points (1, 0), and (b, 1). Step 3 Plot the points and sketch the graph. Example 4

Graph Logarithmic Functions (1, 0) (b, 1) → (3, 1) Answer: Example 4

Step 2 Determine points on the graph. Graph Logarithmic Functions B. Graph the function Step 1 Identify the base. Step 2 Determine points on the graph. Example 4

Graph Logarithmic Functions Step 3 Sketch the graph. Answer: Example 4

A. Graph the function f(x) = log5x. A. B. C. D. Example 4

B. Graph the function . A. B. C. D. Example 4

Concept

This represents a transformation of the graph f(x) = log6 x. Graph Logarithmic Functions This represents a transformation of the graph f(x) = log6 x. ● : The graph is compressed vertically. ● h = 0: There is no horizontal shift. ● k = –1: The graph is translated 1 unit down. Example 5

Graph Logarithmic Functions Answer: Example 5

● |a| = 4: The graph is stretched vertically. Graph Logarithmic Functions ● |a| = 4: The graph is stretched vertically. ● h = –2: The graph is translated 2 units to the left. ● k = 0: There is no vertical shift. Example 5

Graph Logarithmic Functions Answer: Example 5

A. B. C. D. Example 5

A. B. C. D. Example 5

Find Inverses of Exponential Functions A. AIR PRESSURE At Earth’s surface, the air pressure is defined as 1 atmosphere. Pressure decreases by about 20% for each mile of altitude. Atmospheric pressure can be modeled by P = 0.8x, where x measures altitude in miles. Find the atmospheric pressure in atmospheres at an altitude of 8 miles. P = 0.8x Original equation = 0.88 Substitute 8 for x. ≈ 0.168 Use a calculator. Answer: 0.168 atmosphere Example 6

x = 0.8P Replace x with P, replace P with x, and solve for P. Find Inverses of Exponential Functions B. AIR PRESSURE At Earth’s surface, the air pressure is defined as 1 atmosphere. Pressure decreases by about 20% for each mile of altitude. Atmospheric pressure can be modeled by P = 0.8x, where x measures altitude in miles. Write an equation for the inverse of the function. P = 0.8x Original equation x = 0.8P Replace x with P, replace P with x, and solve for P. P = log0.8 x Definition of logarithm Answer: P = log0.8 x Example 6

A. AIR PRESSURE The air pressure of a car tire is 44 lbs/in2 A. AIR PRESSURE The air pressure of a car tire is 44 lbs/in2. The pressure decreases gradually by about 1% for each trip of 50 miles driven. The air pressure can be modeled by P = 44(0.99x), where x measures the number of 50-mile trips. Find the air pressure in pounds per square inch after driving 350 miles. A. 42 B. 41 C. 40 D. 39 Example 6

B. AIR PRESSURE The air pressure of a car tire is 44 lbs/in2 B. AIR PRESSURE The air pressure of a car tire is 44 lbs/in2. The pressure decreases gradually by about 1% for each trip of 50 miles driven. The air pressure can be modeled by P = 44(0.99x), where x measures the number of 50-mile trips. Write an equation for the inverse of the function. A. P = 44 log0.99 x B. P = log0.99 x C. D. Example 6

End of the Lesson