MATH Basic Algebra II  - Unit 20: Natural Logarithms

Natural Exponential and Natural Logarithmic Functions

In the following problems, use a calculator to evaluate the expression rounded to the nearest thousandth.

Hint: Remember since you are rounding to the nearest thousandth, your answers will have three places after the decimal point.

1) Evaluate.

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2)

Evaluate.

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3)

Evaluate.

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4)

Evaluate.

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5)

Evaluate.

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6)

Evaluate.

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In the following problems, state each expression as a single natural logarithm.

7)

Express ln 9 + ln 2 as a single natural logarithm.

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8)

Express 3 ln 5 – ln 25 as a single natural logarithm. 

Hint:  Apply the  Power Property first.

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Solving Equations

In the following problems, solve the equation for x by using the natural logarithm function.  Round the answers to the nearest hundredth.

Hint: Take the natural logarithm of each side of the equation. Then use the properties of natural logarithms to solve for x. Your answer should have two places after the decimal point.

9)

Solve for x.

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10)

Solve for x.

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11)

Solve for x.

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12)

Solve for x.

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13)

Solve for x.

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14)

Solve for x.

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Applications of Natural Logarithms

Hint: Use the continuous compound interest formula, A = pert  , where A is the final amount, p is the principal, r is the interest rate written as a decimal and t is the time. Round answers to the nearest cent, that is two places after the decimal point.

15)

How much will $5000 be worth in 5 years if it is compounded continuously at 3% interest?

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16)

Becky invests $2500 at an annual interest rate of 9%.  How much is the investment worth after 10 years if the interest is compounded continuously?

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17)

Suppose $2000 is invested at 5.75% annual interest compounded continuously for 21 months.  How much will the investment be worth at the end of 21 MONTHS?

Hint: First change 21 months to years.

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18)

If you invest $2400 at an annual interest rate of 4.5% that is compounded continuously, how long will it take for your money to double in value?  Round the answer to the nearest tenth of a year.

Hint: Use $4800 for A

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19)

How long will it take an investment to triple at 8% annual interest that is compounded continuously?

Hint:  Assume that P = $1, and calculate t.  Then, try other amounts for P to see if the initial amount invested changes the time that is needed for an investment to triple at 8%.

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Review

20)

Evaluate.

Click here to review the unit content explanation for Operations with Numbers and Exponents.

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21)

Evaluate.

Click here to review the unit content explanation for Operations with Numbers and Exponents.

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22)

Solve the inequality for x.

Hint: Remember this answer will have two parts.

Click here to review the unit content explanation for Inequalities and Absolute Value Equations.

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23)

Find the inverse of {(–2, 4), (–3, –1), (2, 2), (3, 4)}.  State whether the inverse is a function.

Click here to review the unit content explanation for Functions and Inverses of Functions.

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24)

Find the inverse of the given matrix.

Click here to review the unit content explanation for Solving Systems with Matrix Equations.

25)

Solve the system of equations for x and y. State the answer as an ordered pair.

Hint: You may use either the substitution method or the elimination method.

Click here to review the unit content explanation for Systems of Equations and Linear Inequalities.

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26)

Factor.

Click here to review the unit content explanation for Quadratic Functions.

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27)

Solve for x.

Click here to review the unit content explanation for Quadratic Functions.

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28)

Solve for x.  Express the answer as an exact number. (In other words, express the answer as a fraction rather than a rounded decimal).

Click here to review the content explanation of One-to-One Property of  Logarithms in the unit for Logarithmic Functions.

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29)

Solve for x.  Express the answer as an exact number.  (In other words, express the answer as a fraction rather than a rounded decimal).

Click here to review the content explanation of One-to-One Property of  Logarithms in the unit for Logarithmic Functions.

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30)

Solve for x.

Click here to review the content explanation of One-to-One Property of Logarithms in the unit for Logarithmic Functions.

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31) Extended Learning

Watch the following video, then write a five-sentence paragraph summarizing the video.

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