8­6 Natural Logarithms 2011 May 02, 2011
8­6 Natural Logarithms
Objectives: Evaluate natural logarithmic expressions.
Solve equations using natural logarithms.
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8­6 Natural Logarithms 2011 May 02, 2011
Warm­up
Solve each logarithmic equation.
1. log3 x = 4
2. log16 4 = x
3. log16 x = 4
*Test on FRIDAY!!
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8­6 Natural Logarithms 2011 May 02, 2011
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
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8­6 Natural Logarithms 2011 May 02, 2011
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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8­6 Natural Logarithms 2011 May 02, 2011
INVERSES
The properties of common logarithms apply to natural logarithms as well.
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8­6 Natural Logarithms 2011 May 02, 2011
Simplifying Natural Logarithms
Write 3 ln 6 ­ ln 8 as a single natural logarithm.
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8­6 Natural Logarithms 2011 May 02, 2011
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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8­6 Natural Logarithms 2011 May 02, 2011
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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8­6 Natural Logarithms 2011 May 02, 2011
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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8­6 Natural Logarithms 2011 May 02, 2011
Solving a Natural Logarithmic Equation
Solve ln (3x + 5)2 = 4.
ln (3x + 5)2 = 4
Check your answers:
(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
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8­6 Natural Logarithms 2011 May 02, 2011
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8­6 Natural Logarithms 2011 May 02, 2011
You try! Solve the following problems.
a. ln x = 0.1
( x + 2
3 (
b. ln = 12
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8­6 Natural Logarithms 2011 May 02, 2011
Solving an Exponential Equation
Solve 7e2x + 2.5 = 20.
Slide
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
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8­6 Natural Logarithms 2011 May 02, 2011
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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HOMEWORK
8­6 Natural Logarithms 2011 May 02, 2011
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