8.5 Properties 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

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

1. Product of Power a m a n = a m+n 1. Product Property log b uv = log b u + log b v Proof log b uv = log b (b x b y )= log b b x+y = x + y = log b u + log b v

2. Quotient Property 2. Quotient of Power Proof

3. Power of Power (a m ) n = a mn 3. Power Property log b u t = t log b u Proof log b u t = log b (b x ) t = log b b tx = tx = t log b u

3. Power of Power (a m ) n = a mn 3. Power Property log b u t = t log b u

4. Change-of-Base Formula Proof Note that b x = u, log b u = x Taking the logarithm with base c at both sides: log c b x = log c u orx log c b = log c u

Example 1 Assume that log 9 5 = a, log 9 11 = b, evaluate a)log 9 (5/11) b)log 9 55 c)log d) log 9 (121/45)

Practice A) P. 496 Q 9 – 10 by assuming log 2 7 = a, and log 2 21 = b B) P. 496 Q 14 – 17

Example 2 Expanding the expression a)ln(3y 4 /x 3 ) ln(3y 4 /x 3 ) = ln(3y 4 ) – lnx 3 = ln3 + lny 4 – lnx 3 = ln3 + 4 ln|y|– 3 lnx b) log /6 x 9 log /6 x 9 = log /6 + log 3 x 9 = 5/6 log log 3 x = 5/6 log 3 (3· 2 2 ) + 9 log 3 x = 5/6 (log log ) + 9 log 3 x = 5/6 ( log 3 2) + 9 log 3 x

Practice Expand the expression P. 496 Q 39, 45

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/x 2 ) b) 2 log 3 7 – 5 log 3 x + 6 log 9 y 2 2 log 3 7 – 5 log 3 x + 6 log 9 y 2 = log 3 49 – log 3 x ( log 3 y 2 / log 3 9) = log 3 (49/x 5 ) + 3 log 3 y 2 = log 3 (49y 6 /x 5 )

Practice Condense the expression P. 497 Q

Example 4 Calculate log 4 8 and log 6 15 using common and natural logarithms. a) log 4 8 log 4 8 = log8 / log4 = 3 log2 / (2 log2) = 3/2 log 4 8 = ln8 / ln4 = 3 ln2 / (2 ln2) = 3/2 b) log 6 15 = log15 / log6 = 1.511

Example 5 The Richter magnitude M of an earthquake is based on the intensity I of the earthquake and the intensity I o of an earthquake that can be barely felt. One formula used is M = log(I / I o ). If the intensity of the Los Angeles earthquake in 1994 was times I o, 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 / I o = , M = log(I / I o ) = log = 6.8 I / I o = 100, M = log(I / I o ) = log100 = 2

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

Assignment: 8.4 P496 # Show work 8.5 Properties of Logarithmic