Natural Logarithms
7-6
Vocabulary
Review
Write T for true or F for false.
T
1. The function y 5 logb x, where b . 0 and b 2 1 is called a logarithmic function.
F
2. A logarithmic equation is an equation that contains only one logarithm.
F
3. The logarithm of a power is the difference of the logarithm and the exponent.
Vocabulary Builder
inverse function (noun)
IN
vurs
FUNGK
shun
Definition: To find the inverse function, switch the order of the elements in the
ordered pairs of the function.
Example: function, f(x): {(1, 2), (3, 4)}; inverse function, f 21 (x): {(2, 1), (4, 3)}
Use Your Vocabulary
4. Complete the table of values for the inverse function, f 21 (x), of the function f (x).
f1(x) x 3
f(x) x 3
x
y
Chapter 7
1
0
1
2
2
3
4
6
x
2
3
4
6
y
1
0
1
2
206
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Related Words: function, inverse, input, output
Key Concept Natural Logarithmic Function
If y 5 e x , then x 5 loge y 5 ln y. The natural logarithmic function
is the inverse of y 5 e x , so you can write it as y 5 ln x.
5. If y 5 e 5 , then ln y 5
1
1 y = ex
5 .
2
6. If ln b 5 6, then b 5 e
6
2 y = ln x
.
Problem 1 Simplifying a Natural Logarithmic Expression
Got It? What is ln 7 1 2 ln 5 written as a single natural logarithm?
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7. The expression is simplified below. Write a justification for each step.
ln 7 1 2 ln 5
Write the original expression.
ln 7 1 ln 52
Power Property of Logarithms
ln 7 1 ln 25
Simplify the second term.
ln (7 ? 25)
Product Property of Logarithms
ln 175
Multiply.
Problem 2 Solving a Natural Logarithmic Equation
Got It? What are the solutions of ln x 5 2?
8. Complete: If ln x 5 2, then e
2
5 x.
Got It? What are the solutions of ln (3x 1 5)2 5 4? Check your answers.
9. The equation is solved below. Write a justification for each step.
ln (3x 1 5)2 5 4
Write the original equation.
ln (3x 1 5)2 5 e4
Write in exponential form.
ln 2(03x 1 5 5 4e2
Take the square root of each side.
ln (0 1 523x 5 25 4 e2
Subtract 5 from each side.
ln (0 1 523x 5 (25 4 e2) 4 3
Divide each side by 3.
ln (0 1 523x < 0.7964 or x < 24.1267
Use a calculator.
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Lesson 7-6
10. Check Substitute your values for x in ln (3x 1 5)2 5 4. Use a calculator.
ln Q 3 ?
2
1 5R <
0.7964
4.0001
2
ln Q 3 ? 24.1267 1 5 R <
3.9976
Got It? What are the solutions of ln 2x 1 ln 3 5 2? Check your answers.
11. Circle the property of logarithms that justifies writing ln 2x 1 ln 3 as ln 6x.
Power Property
Product Property
12. Use the simplified equation to solve for x.
Quotient Property
13. Check Substitute your value for x in ln 2x 1 ln 3 5 2.
Use a calculator.
ln 6x 5 2
6x 5 e2
ln 2 ?
1.2315
1 ln 3 <
1.9522
2
x 5 e6
x N 1.2315
Solving an Exponential Equation
Problem 3
Got It? What is the solution of e x22 5 12? Check your answer.
14. Use the justifications at the right to solve the equation.
e x22 5 12
Write the original equation.
ln 12
x5
ln 12
x<
Rewrite in logarithmic form.
12
Add 2 to each side.
4.4849
Use a calculator.
15. Check Substitute your value for x in e x22 5 12. Use a calculator.
e 4.4849 22 < 11.9999
Got It? What is the solution of 2e 2x 5 20? Check your answer.
16. Circle the first step in solving the equation. Underline the second step.
Divide each side of the equation by 2.
Use the Power Property.
Write in exponential form.
Write in logarithmic form.
17. Now solve the equation.
2e2x 5 20
e2x 5 10
2x 5 ln 10
x 5 2ln 10 5 22.3026
18. Check Substitute your value for x in 2e2x 5 20. Use a calculator.
2e 2
Chapter 7
22.3
< 19.9484
208
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x225
Problem 4 Using Natural Logarithms
Got It? Space A spacecraft can attain a stable orbit 300 km above Earth if it reaches
a velocity of 7.7 km/s. The formula for a rocket’s maximum velocity v in kilometers per
second is v 5 20.0098t 1 c ln R. The booster rocket fires for t seconds and the velocity
of the exhaust is c km/s. The ratio of the mass of the rocket filled with fuel to its mass
without fuel is R.
Suppose a booster rocket for a spacecraft has a mass ratio of about 15, an exhaust
velocity of 2.1 km/s, and a firing time of 30 s. Can the spacecraft achieve a stable orbit
300 km above Earth?
19. Identify the values for each variable.
t 5 30
c 5 2.1
R 5 15
20. Complete the equation for finding the spacecraft’s maximum velocity. Then use a
calculator to simplify the equation.
v 5 20.0098 ? 30 1 Q 2.1 ln 15 R km/s < 5.3929 km/s
21. The spacecraft will / will not achieve a stable orbit 300 km above Earth. Explain.
Answers may vary. Sample: The velocity of 5.4 km/s is less than the
_______________________________________________________________________
7.7 km/s needed to attain a stable orbit.
_______________________________________________________________________
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Lesson Check • Do you UNDERSTAND?
Reasoning Can ln 5 1 log2 10 be written as a single logarithm? Explain your reasoning.
22. The ln 5 1 log2 10 can be written as ln 5 5 loge 5
1 log2 10.
23. Can you use the Product Property of Logarithms to combine the logarithms?
Explain why or why not.
No; the two logarithms have different bases.
_______________________________________________________________________
24. ln 5 1 log2 10 can / cannot be written as a single logarithm.
Math Success
Check off the vocabulary words that you understand.
function
logarithm
natural logarithmic function
Rate how well you can write and solve equations with natural logarithms.
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Lesson 7-6