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86 Natural Logarithms 2011
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May 02, 2011
86 Natural Logarithms
Objectives: Evaluate natural logarithmic expressions.Solve equations using natural logarithms.
86 Natural Logarithms 2011
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WarmupSolve each logarithmic equation.
1. log3 x = 4 2. log16 4 = x 3. log16 x = 4
*Test on FRIDAY!!
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Check Skills You'll Need
Use your calculator to evaluate each expression to the nearest thousandth.
1. e5 2. 2e3 3. e2 4. 1/e 5. 4.2e
Solve.
6. log3 x = 4 7. log16 4 = x 8. log16 x = 4
Use the properties of logarithms to evaluate each expression.
9. log2 8 log2 4 10. ¾ log 10 + 7 log 100 11. 7 log8 49
86 Natural Logarithms 2011
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In lesson 82 you learned that e ≈ 2.71828.
A logarithm that has a base of e has a special name called a
NATURAL LOGARITHM.
Instead of writing loge x, we now write natural logarithms like this: ln x
Therefore,
loge x = ln x
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The properties of common logarithms apply to natural logarithms as well.
INVERSES
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Simplifying Natural Logarithms
Write 3 ln 6 ln 8 as a single natural logarithm.
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Try some more!
a. 5 ln 2 ln 4 b. 3 ln x + ln y c. ¼ ln 3 + ¼ ln x
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With a partner, answer this question:
What is ln e?
loge e = x
ex = e
x must be 1
Therefore, ln e = 1
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There is a special button on your calculator for natural logarithms. Use that button to evaluate the following to four decimal places:
a. ln 4
b. 7 ln 3.2
c. ½ ln 2 2 ln 5
d. ln (6)
Why can't you take the natural log of a negative number?
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Solving a Natural Logarithmic Equation
Solve ln (3x + 5)2 = 4.
ln (3x + 5)2 = 4
(3x + 5)2 = e4
(3x + 5)2 = 54.6
3x + 5 = ±√54.6
3x + 5 = 7.39 or 7.39
x = 0.797 or x = 4.13
Check your answers:
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You try! Solve the following problems.
a. ln x = 0.1 b. ln = 12x + 23( (
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Solving an Exponential EquationSolve 7e2x + 2.5 = 20.
7e2x + 2.5 = 20
7e2x = 17.5
e2x = 2.5
ln e2x = ln 2.5
2x ln e = ln 2.5
2x = ln 2.5
x = (ln 2.5)/2
x ≈ 0.458
Slide
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You would like to go to the Master's Tournament. Right now you have $1500. If you invest this money in an account that compounds continuously at a rate of 6.5%, how many years will it take for you to have $4000 in your account so you have enough money to go to the tournament?
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