1. 8-6 Natural Logarithms

2. Objectives
Natural Logarithms Natural Logarithmic & Exponential Equations

3. the x comes down from the exponent.
Vocabulary
y = ex and y = ln x are inverses So, e and ln are inverse operations. Ex. ln x = 4 ex = 12 elnx = e4 ln (ex) = ln 12 x = e4 x = ln 12 x = 54.6 x = 2.48
Cancel each other
the x comes down from the exponent.

4. Simplify Natural Logarithms
Write 2 ln 12 – ln 9 as a single natural logarithm.
2 ln 12 – ln 9 = ln 122 – ln 9 Power Property
= ln Quotient Property
122 9
= ln 16 Simplify.

5. Real World Example
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) ln 22 Substitute.
– (3.091) Use a calculator.
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.

6. Solving a Natural Logarithm Equation
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 + 4 2
x Use a calculator.
Check: ln (2 • 5.69 – 4) ln

7. Solving an Exponential Equation
Use natural logarithms to solve 4e3x = 14.
4e3x = 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 3.2 3 x

8. Real World Example
An initial investment of $200 is now valued at $ The interest rate is 6%, compounded continuously. How long has the money been invested?
A = Pert Continuously compounded interest formula.
= 200e0.06t Substitute for A, 200 for P, and 0.06 for r.
= e0.06t Divide each side by 200.
ln = ln e0.06t Take the natural logarithm of each side.
ln = 0.06t Simplify.
= t Solve for t. ln
0.06
4 t Use a calculator.
The money has been invested for 4 years.

9. Homework
8-6 p ,2,10,11,14,15,23,24