• 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
A. What is ex = 15 in logarithmic form?
A. ln e = 15
B. ln 15 = e
0%
A
B
C
0%
D
D
A
0%
B
D. ln 15 = x
C
C. ln x = 15
A.
B.
C.
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D.
Write Equivalent Expressions
B. Write an equivalent logarithmic equation for
e4 = x.
e4 = x → loge x = 4
ln x = 4
Answer: ln x = 4
B. What is e4 = x in logarithmic form?
A. ln e = 4
B. ln x = 4
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A
B
C
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D
D
A
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B
D. ln 4 = x
C
C. ln x = e
A.
B.
C.
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D.
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
A. Write ln x ≈ 1.5763 in exponential form.
A. x ≈ 1.5763e
B. x ≈ e1.5763
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B
A
0%
A
B
C
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D
D
D. e ≈ 1.5763x
C
C. e ≈ x1.5763
A.
B.
C.
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D.
Write Equivalent Expressions
B. Write ln 25 ≈ x in exponential form.
ln 25 ≈ x → loge 25 = x
25 ≈ ex
Answer: 25 ≈ ex
B. Write ln 47 = x in exponential form.
A. 47 = ex
B. e = 47x
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B
A
0%
A
B
C
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D
D
D. 47 = xe
C
C. x = 47e
A.
B.
C.
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D.
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
= ln 486
Answer: ln 486
6) Product Property of Logarithms
Simplify.
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
= ln (32
= ln 36y
Answer: ln 36y
4
y)
Power Property
of Logarithms
Product
Property of
Logarithms
Simplify.
A. Write 4 ln 2 + In 3 as a single logarithm.
A. ln 6
B. ln 24
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B
A
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A
B
C
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D
D
D. ln 48
C
C. ln 32
A.
B.
C.
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D.
1 + ln x as a single logarithm.
B. Write 3 ln 3 + ln __
3
A. ln 3x
B. ln 9x
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B
A
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A
B
C
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D
D
D. ln 27x
C
C. ln 18x
A.
B.
C.
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D.
Solve Base e Equations
Solve 3e–2x + 4 = 10. Round to the nearest
ten-thousandth.
3e–2x + 4 = 10
3e–2x = 6
Original equation
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.
What is the solution to the equation 2e–2x + 5 = 15?
A. –0.8047
B. –0.6931
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B
A
0%
A
B
C
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D
D
D. 0.8047
C
C. 0.6931
A.
B.
C.
0%
D.
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.
A. SAVINGS Suppose you deposit $700 into an
account paying 6% annual interest, compounded
continuously. What is the balance after 7 years?
A. $46,058.59
B. $46,680.43
C. $1065.37
D. $365.37
A.
B.
C.
D.
A
B
C
D