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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. ln e = 15
B. ln 15 = e
C. ln x = 15
D. ln 15 = x
A. What is ex = 15 in logarithmic form?
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. 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. 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. ln 6
B. ln 24
C. ln 32
D. ln 48
A. Write 4 ln 2 + In 3 as a single logarithm.
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. –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.
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
(3x+1)² > e8 Write in exponential form
(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. 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. x > 274.66
B. x > 282.84
C. x > 286.91
D. x < 294.85
B. Solve the inequality in (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.
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.
Homework
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. 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. $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?
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