Five-Minute Check (over Lesson 8–6)

Then/Now

New Vocabulary

Key Concept: Natural Base Functions

Example 1: Write Equivalent Expressions

Example 2: Write Equivalent Expressions

Example 3: Simplify Expressions with e and the Natural Log

Example 4: Solve Base e Equations

Example 5: Solve Natural Log Equations and Inequalities

Key Concept: Continuously Compounded Interest

Example 6: Real-World Example: Solve Base e Inequalities

Over Lesson 8–6

A. A

B. B

C. C

D. D0% 0%0%0%

A. 0.5315

B. 1.5314

C. 1.2238

D. 29.9641

Use a calculator to evaluate log 3.4 to the nearest ten-thousandth.

Over Lesson 8–6

A. A

B. B

C. C

D. D0% 0%0%0%

A. 6.7256

B. 7.0164

C. 7.8074

D. 9.2381

Solve 2x – 4 = 14. Round to the nearest ten-thousandth.

Over Lesson 8–6

A. A

B. B

C. C

D. D0% 0%0%0%

A. {p | p = 4}

B. {p | p > 3.6998}

C. {p | p < 3.4679}

D. {p | p > 2.5713}

Solve 42p – 1 > 11p. Round to the nearest ten-thousandth.

Over Lesson 8–6

A. A

B. B

C. C

D. D0% 0%0%0%

A. –3.4829

B. 1.5

C. 1.6845

D. 1.7063

Express log4 (2.2)3 in terms of common logarithms. Then approximate its value to four decimal places.

Over Lesson 8–6

A. A

B. B

C. C

D. D0% 0%0%0%

Solve for x: 92x = 45.

A.

B.

C.

D. x = 2 log 5

You worked with common logarithms. (Lesson 8–6)

• Evaluate expressions involving the natural base and natural logarithm.

• Solve exponential equations and inequalities using natural logarithms.

• natural base, e

• natural base exponential function

• natural logarithm

Write Equivalent Expressions

A. Write an equivalent logarithmic equation for ex = 23.

ex = 23 → loge 23 = x

ln 23 = x

Answer: ln 23 = x

Write Equivalent Expressions

B. Write an equivalent logarithmic equation for e4 = x.

e4 = x → loge x = 4

ln x = 4

Answer: ln x = 4

A. A

B. B

C. C

D. D0% 0%0%0%

A. ln e = 15

B. ln 15 = e

C. ln x = 15

D. ln 15 = x

A. What is ex = 15 in logarithmic form?

A. A

B. B

C. C

D. D0% 0%0%0%

A. ln e = 4

B. ln x = 4

C. ln x = e

D. ln 4 = x

B. What is e4 = x in logarithmic form?

Write Equivalent Expressions

A. Write ln x ≈ 1.2528 in exponential form.

ln x ≈ 1.2528 → loge x = 1.2528

x ≈ e1.2528

Answer: x ≈ e1.2528

Write Equivalent Expressions

B. Write ln 25 ≈ x in exponential form.

ln 25 ≈ x → loge 25 = x

25 ≈ ex

Answer: 25 ≈ ex

A. A

B. B

C. C

D. D0% 0%0%0%

A. x ≈ 1.5763e

B. x ≈ e1.5763

C. e ≈ x1.5763

D. e ≈ 1.5763x

A. Write ln x ≈ 1.5763 in exponential form.

A. A

B. B

C. C

D. D0% 0%0%0%

A. 47 = ex

B. e = 47x

C. x = 47e

D. 47 = xe

B. Write ln 47 = x in exponential form.

Simplify Expressions with e and the Natural Log

A. Write 4 ln 3 + ln 6 as a single algorithm.

4 ln 3 + ln 6 = ln 34 + ln 6 Power Property of Logarithms

= ln (34 ● 6) Product Property of Logarithms

= ln 486 Simplify.Answer: ln 486

Simplify Expressions with e and the Natural Log

Check Use a calculator to verify the solution.

4 3 6LN ENTER) + LN

Keystrokes:

)

486 6.1862 LN ENTER)

Keystrokes:

Simplify Expressions with e and the Natural Log

B. Write 2 ln 3 + ln 4 + ln y as a single algorithm.

2 ln 3 + ln 4 + ln y = ln 32 + ln 4 + ln y Power Property of Logarithms

= ln (32 ● 4 ● y) Product Property of Logarithms

= ln 36y Simplify.Answer: ln 36y

A. A

B. B

C. C

D. D0% 0%0%0%

A. ln 6

B. ln 24

C. ln 32

D. ln 48

A. Write 4 ln 2 + In 3 as a single logarithm.

A. A

B. B

C. C

D. D0% 0%0%0%

A. ln 3x

B. ln 9x

C. ln 18x

D. ln 27x

B. Write 3 ln 3 + ln + ln x as a single logarithm.__13

Solve Base e Equations

Solve 3e–2x + 4 = 10. Round to the nearest ten-thousandth.

3e–2x + 4= 10Original equation

3e–2x = 6 Subtract 4 from each side.

e–2x = 2 Divide each side by 3.

ln e–2x = ln 2 Property of Equality for Logarithms

–2x = ln 2 Inverse Property of Exponents and Logarithms

Divide each side by –2.

Solve Base e Equations

x≈ –0.3466 Use a calculator.

Answer: The solution is about –0.3466.

A. A

B. B

C. C

D. D0% 0%0%0%

A. –0.8047

B. –0.6931

C. 0.6931

D. 0.8047

What is the solution to the equation 2e–2x + 5 = 15?

Solve Natural Log Equations and Inequalities

A. Solve 2 ln 5x = 6. Round to the nearest ten-thousandth.

Answer: about 4.0171

2 ln 5x = 6 Original equation

ln 5x = 3 Divide each side by 2.

eln 5x = e3 Property of Equality for Exponential Functions

5x = e3 eln x = x

Divide each side by 5.

x ≈ 4.0171 Use a calculator.

Solve Natural Log Equations and Inequalities

B. Solve the inequality ln (3x + 1)2 > 8. Round to the nearest ten-thousandth.

ln (3x + 1)2 > 8 Original equation

eln (3x + 1)2 > e8 Write each side using exponents and base e.

(3x + 1)2 > (e4)2 eln x = x and Power of of Power

3x + 1 > e4 Property of Inequality for Exponential Functions

3x > e4 – 1 Subtract 1 from each side.

Solve Natural Log Equations and Inequalities

x > 17.8661 Use a calculator.

Divide each side by 3.

Answer: x > 17.8661

A. A

B. B

C. C

D. D0% 0%0%0%

A. 7.8732

B. 8.0349

C. 9.0997

D. 11.232

A. Solve the equation 3 ln 6x = 12. Round to the nearest ten-thousandth.

A. A

B. B

C. C

D. D0% 0%0%0%

A. x > 274.66

B. x > 282.84

C. x > 286.91

D. x < 294.85

B. Solve the inequality ln (4x –2) > 7. Round to the nearest ten-thousandth.

Solve Base e Inequalities

A. SAVINGS Suppose you deposit $700 into an account paying 3% annual interest, compounded continuously. What is the balance after 8 years?

A = Pert Continuously Compounded Interest formula

= 700e(0.03)(8)

Replace P with 700, r with 0.03 and t with 8.

= 700e0.24 Simplify.

≈ 889.87 Use a calculator.Answer: The balance after 8 years will be $889.87.

Solve Base e Inequalities

B. SAVINGS Suppose you deposit $700 into an account paying 3% annual interest, compounded continuously. How long will it take for the balance in your account to reach at least $1200?

The balance is at least $1200.

A ≥ 1200 Write an inequality.

Replace A with 700e(0.03)t.

Divide each side by 700.

Solve Base e Inequalities

Answer: It will take about 18 years for the balance to reach at least $1200.

Property of Inequality for Logarithms

Inverse Property of Exponents and Logarithms

Divide each side by 0.03.

t ≥ 17.97 Use a calculator.

Solve Base e Inequalities

C. SAVINGS Suppose you deposit $700 into an account paying 3% annual interest, compounded continuously. How much would have to be deposited in order to reach a balance of $1500 after 12 years?

A = Pert Continuously Compounded Interest formula

1500= P ● e0.03 ● 12

A = 1500, r = 0.003, and t = 12

Divide each side by e0.36.

Solve Base e Inequalities

1046.51≈ P Use a calculator.

Answer: You need to deposit $1046.51.

A. A

B. B

C. C

D. D0% 0%0%0%

A. $46,058.59

B. $46,680.43

C. $1065.37

D. $365.37

A. SAVINGS Suppose you deposit $700 into an account paying 6% annual interest, compounded continuously. What is the balance after 7 years?

A. A

B. B

C. C

D. D0% 0%0%0%

A. at least 1.27 years

B. at least 7.50 years

C. at least 21.22 years

D. at least 124.93 years

B. SAVINGS Suppose you deposit $700 into an account paying 6% annual interest, compounded continuously. How long will it take for the balance in your account to reach at least $2500?

A. A

B. B

C. C

D. D0% 0%0%0%

A. $1299.43

B. $1332.75

C. $1365.87

D. $1444.60

C. SAVINGS Suppose you deposit money into an account paying 3% annual interest, compounded continuously. How much would have to be deposited in order to reach a balance of $1950 after 10 years?