4.3 Rules of Logarithms

# 4.3 Rules of Logarithms

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## 4.3 Rules of Logarithms

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1. 4.3 Rules of Logarithms Objectives: 1. Compare & recall the properties of exponents 2. Deduce the properties of logarithms from/by comparing the properties of exponents 3. Use the properties of logarithms 4. Application Vocabulary: change-of-base formula

2. Pre-Knowledge For any b, c, u, v R+, and b ≠ 1, c ≠ 1, there exists some x, y  R, such that u = bx, v = by By the previous section knowledge, as long as taking x = logbu, y = logbv

3. 1. Product of Power am an = am+n 1. Product Property logbuv = logbu + logbv Proof logbuv = logb(bxby)= logbbx+y = x + y = logbu + logbv

4. 2. Quotient of Power 2. Quotient Property Proof

5. 3. Power of Power (am)n = amn 3. Power Property logbut = t logbu Proof logbut = logb(bx)t = logbbtx = tx = t logbu

6. 3. Power of Power (am)n = amn 3. Power Property logbut = t logbu

7. 4. Change-of-Base Formula Proof Note that bx = u, logbu = x Taking the logarithm with base c at both sides: logcbx = logcu or x logcb = logcu

8. Example 1 Assume that log95 = a, log911 = b, evaluate • log9 (5/11) • log955 • log9125 d) log9(121/45)

9. Practice A) by assuming log27 = a, and log221 = b B)

10. Example 2 Expanding the expression • ln(3y4/x3) ln(3y4/x3) = ln(3y4)– lnx3 = ln3 + lny4 – lnx3 = ln3 + 4 ln|y|– 3 lnx b) log3125/6x9 log3125/6x9 = log3125/6 + log3x9 = 5/6 log312 + 9 log3x = 5/6 log3(3· 22) + 9 log3x = 5/6 (log33 + log322) + 9 log3x = 5/6 ( 1 + 2 log32) + 9 log3x

11. Practice Expand the expression

12. Example 3 Condensing the expression a) 3 ( ln3 – lnx ) + ( lnx – ln9 ) 3 ( ln3 – lnx ) + ( lnx – ln9 ) = 3 ln3 – 3 lnx + lnx – 2 ln3 = ln3 – 2 lnx = ln(3/x2) b) 2 log37 – 5 log3x + 6 log9y2 2 log37 – 5 log3x + 6 log9y2 = log349 – log3x5 + 6 ( log3y2/ log39) = log3(49/x5) + 3 log3y2 = log3(49y2/x5)

13. Practice Condense the expression

14. Example 4 Calculate log48 and log615 using common and natural logarithms. a) log48 log48 = log8 / log4 = 3 log2 / (2 log2) = 3/2 log48 = ln8 / ln4 = 3 ln2 / (2 ln2) = 3/2 b) log615 = log15 / log6 = 1.511

15. Example 5The Richter magnitude M of an earthquake is based on the intensity I of the earthquake and the intensity Io of an earthquake that can be barely felt. One formula used is M = log(I / Io). If the intensity of the Los Angeles earthquake in 1994 was 106.8 times Io, what was the magnitude of the earthquake? What magnitude on the Richter scale does an earthquake have if its intensity is 100 times the intensity of a barely felt earthquake? I / Io = 106.8, M = log(I / Io) = log106.8 = 6.8 I / Io = 100, M = log(I / Io) = log100 = 2

16. Challenge Simplify (No calculator) 1) 2) 3) 4) 5) Proof

17. Assignment: