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Algebra 2Algebra 2
Write 2 ln 12 – ln 9 as a single natural logarithm.
2 ln 12 – ln 9 = ln 122 – ln 9 Power Property
= ln Quotient Property122
9
= ln 16 Simplify.
Lesson 8-6
Natural LogarithmsNatural Logarithms
Additional Examples
Algebra 2Algebra 2
Find the velocity of a spacecraft whose booster rocket has a
mass ratio 22, an exhaust velocity of 2.3 km/s, and a firing time of 50 s.
Can the spacecraft achieve a stable orbit 300 km above Earth?
Let R = 22, c = 2.3, and t = 50. Find v.
v = –0.0098t + c ln R Use the formula.
= –0.0098(50) + 2.3 ln 22 Substitute.
–0.49 + 2.3(3.091) Use a calculator.
6.62 Simplify.
The velocity is 6.6 km/s is less than the 7.7 km/s needed for a stable orbit. Therefore, the spacecraft cannot achieve a stable orbit at 300 km above Earth.
Lesson 8-6
Natural LogarithmsNatural Logarithms
Additional Examples
Algebra 2Algebra 2
Solve ln (2x – 4)3 = 6.
ln (2x – 4)3 = 6
3 ln (2x – 4) = 6 Power Property
ln (2x – 4) = 2 Divide each side by 3.
2x – 4 = e2 Rewrite in exponential form.
x = Solve for x.e2 + 42
x 5.69 Use a calculator.
Check: ln (2 • 5.69 – 4)3 6 ln 401.95 6 5.996 6
Lesson 8-6
Natural LogarithmsNatural Logarithms
Additional Examples
Algebra 2Algebra 2
Use natural logarithms to solve 4e3x + 1.2 = 14.
4e3x + 1.2 = 14
4e3x = 12.8 Subtract 1.2 from each side.
e3x = 3.2 Divide each side by 4.
ln e3x = ln 3.2 Take the natural logarithm of each side.
3x = ln 3.2 Simplify.
x = Solve for x.ln 3.23
x 0.388
Lesson 8-6
Natural LogarithmsNatural Logarithms
Additional Examples
Algebra 2Algebra 2
An initial investment of $200 is now valued at $254.25. The
interest rate is 6%, compounded continuously. How long has the
money been invested?
A = Pert Continuously compounded interest formula.
254.25 = 200e0.06t Substitute 254.25 for A, 200 for P, and 0.06 for r.
1.27125 = e0.06t Divide each side by 200.
ln 1.27125 = ln e0.06t Take the natural logarithm of each side.
ln 1.27125 = 0.06t Simplify.
The money has been invested for 4 years.
= t Solve for t.ln 1.271250.06
4 t Use a calculator.
Lesson 8-6
Natural LogarithmsNatural Logarithms
Additional Examples
