Name
LESSON
Date
Class
Reteach
2⫺2
7-6 The Natural Base, e
The natural logarithmic function, f x ⫽ ln x, is the inverse of the exponential function with
the natural base e, f x ⫽ e x.
The constant e is an irrational number. e 2.71828….
Properties of logarithms apply to the natural logarithm.
In particular:
ln 1 ⫽ 0
0
The base is e and e ⫽ 1.
ln e ⫽ 1
1
Think: e ⫽ e.
x
ln e ⫽ x
ln x
e
⫽x
The natural logarithm and the
exponential function are inverses,
so they undo each other.
Use properties of logarithms to simplify expressions with e or “ln.”
Simplify: ln e x ⫹ 2
Simplify: e 4 ln x
Step 1 Use the Power Property.
“Bring down” the exponent
to multiply.
Step 1 Use the Power Property.
Write the exponent.
ln e x ⫹ 2
x ⫹ 2 ln e
e
4 ln x
ln x 4
ln e ⫽ 1
e
Step 2 Simplify.
e ln x
Step 2 Simplify.
x ⫹ 2 ln e
x⫹2
e ln x ⫽ x
4
x4
Simplify each expression.
1. ln e ⫺6x
2. ln e t ⫺ 3
⫺6x ln e
3. e 2 ln x
t ⫺ 3 ln e
e ln x
2
t⫺3
⫺6x
5. ln e x ⫹ 1
4. ln e 1.8
1.8 ln e
x
6. e 7 ln x
⫹ 1 ln e
x⫹1
1.8
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
a207c07-6_rt.indd 46
x2
46
e
ln x 7
x
7
Holt Algebra 2
1/4/06 3:39:28 PM
Process Black
Name
LESSON
Date
Class
Reteach
2⫺2
7-6 The Natural Base, e (continued)
The natural base, e, appears in the formula for interest compounded continuously.
A ⫽ Pe
rt
A ⫽ total amount
P ⫽ principal, or initial amount
r ⫽ annual interest rate
t ⫽ time in years
What is the total amount for an investment of $2000 invested at 3% and compounded
continuously for 5 years?
Step 1
Identify the values that correspond to the variables in the formula.
P ⫽ initial investment ⫽ $2000
r ⫽ 3% ⫽ 0.03
t⫽5
Step 2
Substitute the known values into the formula.
A ⫽ Pe
rt
A ⫽ 2000e
Step 3
0.03 5 Use a calculator to solve for A, the total amount.
A ⫽ 2000e
0.03 5 Use the e x key on a calculator:
2000e .03*5 ⫽ 2323.668485
A " 2323.67
The total amount is $2323.67.
rt
Use the formula A ⫽ Pe to solve.
7. What is the total amount for an investment of $500 invested at 4.5% and
compounded continuously for 10 years?
P⫽
500
r⫽
0.045
t⫽
10
$784.16
8. Randy deposited $1000 into an account that paid
2.8% with continuous compounding. What was her
balance after 6 years?
$1182.94
9. a. Martin borrows $5500. The rate is set at 6% with
continuous compounding. How much does he
owe at the end of 2 years?
$6201.23
b. Martin found a bank with a better interest rate
of 5.5%. How much less does he owe at the
end of 2 years?
$61.70
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
a207c07-6_rt.indd 47
47
Holt Algebra 2
1/4/06 3:39:29 PM
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AK4up.indd 78
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Holt Algebra 2
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