Transcript 8-6 Natural Logarithms - Mr. Hale's Classes
8-6 Natural Logarithms
Objectives
Natural Logarithms Natural Logarithmic & Exponential Equations
Vocabulary
y =
e
x and y = ln x are inverses So,
e
and ln are inverse operations.
Ex. ln x = 4 e lnx = e 4 x = e 4 e ln (e x x = 12 ) = ln 12 x = ln 12 x = 54.6 x = 2.48
Cancel each other the x comes down from the exponent.
Simplify Natural Logarithms
Write 2 ln 12 – ln 9 as a single natural logarithm.
2 ln 12 – ln 9 = ln 12 2 – ln 9 Power Property = ln 12 2 9 = ln 16 Quotient Property Simplify.
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.
v
= –0.0098
t
+
c
ln
R
= –0.0098( 50 ) + 2.3
ln 22 –0.49 + 2.3(3.091) 6.62
Find
v
.
Use the formula.
Substitute.
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.
Solving a Natural Logarithm Equation
Solve ln (2
x
– 4) 3 = 6.
ln (2
x
– 4) 3 = 6 3 ln (2
x
– 4) = 6 ln (2
x
– 4) = 2 2
x
– 4 =
e
2
x x
=
e
2 2 + 4 5.69
Power Property Divide each side by 3.
Rewrite in exponential form.
Solve for
x
.
Use a calculator.
Check:
ln (2 • 5.69
– 4) 3 6 ln 401.95 6 5.996 6
Solving an Exponential Equation
Use natural logarithms to solve 4
e
3
x
+ 1.2 = 14.
4
e
3
x
+ 1.2 = 14 4
e
3
x
= 12.8
Subtract 1.2 from each side.
e
3
x
= 3.2
ln
e
3
x
= ln 3.2
Divide each side by 4.
Take the natural logarithm of each side.
3
x
= ln 3.2
x
= ln 3.2
3 Simplify.
Solve for
x
.
x
0.388
Real World Example
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
=
Pe rt
254.25
= 200
e
0.06
t
1.27125 =
e
0.06
t
ln 1.27125 = ln
e
0.06
t
ln 1.27125 = 0.06
t
ln 1.27125
=
t
0.06
4
t
Continuously compounded interest formula.
Substitute 254.25 for
A
, 200 for
P
, and 0.06 for
r
.
Divide each side by 200.
Take the natural logarithm of each side.
Simplify.
Solve for
t
.
Use a calculator.
The money has been invested for 4 years.
Homework
8-6 p 472 1,2,10,11,14,15,23,24