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Properties of Logarithms Section 3.3 Objectives • Rewrite logarithms with different bases. • Use properties of logarithms to evaluate or rewrite logarithmic expressions. • Use properties of logarithms to expand or condense logarithmic expressions. History of Logarithms John Napier, a 16th Century Scottish scholar, contributed a host of mathematical discoveries. John Napier (1550 – 1617) He is credited with creating the first computing machine, logarithms and was the first to describe the systematic use of the decimal point. Other contributions include a mnemonic for formulas used in solving spherical triangles and two formulas known as Napier's analogies. “In computing tables, these large numbers may again be made still larger by placing a period after the number and adding ciphers. ... In numbers distinguished thus by a period in their midst, whatever is written after the period is a fraction, the denominator of which is unity with as many ciphers after it as there are figures after the period.” Napier lived during a time when revolutionary astronomical discoveries were being made. Copernicus’ theory of the solar system was published in 1543, and soon astronomers were calculating planetary positions using his ideas. But 16th century arithmetic was barely up to the task and Napier became interested in this problem. Nicolaus Copernicus (1473-1543) Even the most basic astronomical arithmetic calculations are ponderous. Johannes Kepler (1571-1630) filled nearly 1000 large pages with dense arithmetic while discovering his laws of planetary motion! A typical page from one of Kepler’s notebooks Johannes Kepler (1571-1630) Napier’s Bones In 1617, the last year of his life, Napier invented a tool called “Napier's Bones” which reduces the effort it takes to multiply numbers. “Seeing there is nothing that is so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications, divisions... I began therefore to consider in my mind by what certain and ready art I might remove those hindrances.” Logarithms Appear The first definition of the logarithm was constructed by Napier and popularized by a pamphlet published in 1614, two years before his death. His goal: reduce multiplication, division, and root extraction to simple addition and subtraction. Napier defined the "logarithm" L of a number N by: N==107(1-10(-7))L This is written as NapLog(N) = L or NL(N) = L While Napier's definition for logarithms is different from the modern one, it transforms multiplication and division into addition and subtraction in exactly the same way. Logarithmic FAQs • Logarithms are a mathematical tool originally invented to reduce arithmetic computations. • Multiplication and division are reduced to simple addition and subtraction. • Exponentiation and root operations are reduced to more simple exponent multiplication or division. • Changing the base of numbers is simplified. • Scientific and graphing calculators provide logarithm functions for base 10 (common) and base e (natural) logs. Both log types can be used for ordinary calculations. Logarithmic Notation • For logarithmic functions we use the notation: loga(x) or logax • This is read “log, base a, of x.” Thus, y = logax means x = ay • And so a logarithm is simply an exponent of some base. Change-of-Base Formula Only logarithms with base 10 or base e can be found by using a calculator. Other bases require the use of the Change-of-Base Formula. Change-of-Base Formula If a 1, and b 1, and M are positive real numbers, then logb M log a M . logb a Example: Approximate log4 25. log4 25 10 is used for both bases. log10 25 log 25 1.39794 2.32193 log10 4 log 4 0.60206 The Change-of-Base Rule Change-of-Base Rule For any positive real numbers x, a, and b, where a 1 and b 1, log x log b x . a log b a Proof Let y log a x. ay x log b a y log b x y log b a log b x log y log x ba b log log a x log x ba b Change of base formula: • u, b, and c are positive numbers with b≠1 and c≠1. Then: log b u • logcu = log b c log u • logcu = log c (base 10) ln u • logcu = ln c (base e) Change-of-Base Formula Example: Approximate the following logarithms. (a) log3 198 log3 198 log198 2.297 4.816 log 3 0.477 (b) log6 5 log6 log 5 0.349 0.449 5 0.778 log 6 Examples: • Use the change of base to evaluate: • log37 = • (base 10) •(base e) • log 7 ≈ • log 3 • 1.771 •ln 7 ≈ •ln 3 •1.771 Your Turn: Evaluate each expression and round to four decimal places. (a) log 5 17 Solution (a) 1.7604 (b) -3.3219 (b) log 2 .1 Properties of Logarithms For x > 0, y > 0, a > 0, a 1, and any real number r, Product Rule log a xy log a x log a y. Quotient Rule log a xy log a x log a y. r log x r log a x. Power Rule a Examples Assume all variables are positive. Rewrite each expression using the properties of logarithms. 1. log 8 x log 8 log x 15 2. log9 log9 15 log9 7 7 1 1 2 3. log5 8 log5 8 log5 8 2 The Product Rule of Logarithms Product Rule of Logarithms If M, N, and a are positive real numbers, with a 1, then loga(MN) = logaM + logaN. Example: Write the following logarithm as a sum of logarithms. (a) log5(4 · 7) log5(4 · 7) = log54 + log57 (b) log10(100 · 1000) log10(100 · 1000) = log10100 + log101000 =2+3=5 Your Turn: • Express as a sum of logarithms: 2 log3 ( x w) Solution: log3 ( x w) log3 x log3 w 2 2 The Quotient Rule of Logarithms Quotient Rule of Logarithms If M, N, and a are positive real numbers, with a 1, then log M log M log N. a N a a Example: Write the following logarithm as a difference of logarithms. 10 (a) log5 = log5 10 log5 3 3 c (b) log8 log8 c log8 4 4 Your Turn: • Express as a difference of logarithms. 10 log a b • Solution: 10 log a log a 10 log a b b Sum and Difference of Logarithms 8y log Example: Write as the sum or difference 6 5 of logarithms. 8y log6 log 6(8 y) log 6 5 Quotient Rule 5 log 6 8 log 6 y log 6 5 Product Rule The Power Rule of Logarithms The Power Rule of Logarithms If M and a are positive real numbers, with a 1, and r is any real number, then loga M r = r loga M. Example: Use the Power Rule to express all powers as factors. log4(a3b5) = log4(a3) + log4(b5) = 3 log4a + 5 log4b Product Rule Power Rule Your Turn: • Express as a product. log a 7 3 Solution: 3 log a 7 3log a 7 Your Turn: • Express as a product. 5 log a 11 • Solution: log a 11 log a 11 5 1/5 1 log a 11 5 NOT Laws of Logarithms Warning Rewriting Logarithmic Expressions • The properties of logarithms are useful for rewriting logarithmic expressions in forms that simplify the operations of algebra. • This is because the properties convert more complicated products, quotients, and exponential forms into simpler sums, differences, and products. • This is called expanding a logarithmic expression. • The procedure above can be reversed to produce a single logarithmic expression. • This is called condensing a logarithmic expression. Examples: • Expand: • log 5mn = • log 5 + log m + log n • Expand: • log58x3 = • log58 + 3·log5x Expand – Express as a Summ and Difference of Logarithms 7x • log2 = y 3 • log27x3 - log2y = • log27 + log2x3 – log2y = • log27 + 3·log2x – log2y Condense - Express as a Single Logarithm Example: Write the following as the logarithm of a single expression. 5log6(x 3) [2log6(x 4) 3log 6 x] 5log6(x 3) [2log6(x 4) 3log 6 x] log 6(x 3)5 [log 6(x 4) 2 log 6 x3] Power Rule log 6(x 3)5 [log 6(x 4) 2 x3] Product Rule (x 3)5 log 6 2 3 ( x 4) x Quotient Rule Condensing Logarithms • log 6 + 2 log2 – log 3 = • log 6 + log 22 – log 3 = • log (6·22) – log 3 = • log 6 2 = 2 3 • log 8 Examples: • Condense: • log57 + 3·log5t = • log57t3 • Condense: • 3log2x – (log24 + log2y)= 3 x • log2 4y Your Turn: • Express in terms of sums and differences of logarithms. 3 wy log a 2 z 4 • Solution: 3 4 w y 3 4 2 log a 2 log a ( w y ) log a z z log a w3 log a y 4 log a z 2 3log a w 4log a y 2log a z Your Turn: Assume all variables are positive. Use the properties of logarithms to rewrite the expression log 3 5 n x y b zm . Solution logb n x y x y log b m m z z 3 5 3 5 1 n 3 5 x y 1 logb m n z 1 logb x 3 logb y 5 logb z m n 1 3 logb x 5 logb y m logb z n 3 logb x 5 logb y m logb z n n n Your Turn: • Express as a single logarithm. 1 6log b x 2log b y log b z 3 • Solution: 1 6logb x 2logb y logb z logb x 6 logb y 2 logb z1/3 3 x6 1/3 logb 2 log b z y x 6 z1/3 x6 3 z logb 2 , or log b 2 y y Your Turn: Use the properties of logarithms to write 2 3 1 log m log 2 n log m n as a single b b b 2 2 logarithm with coefficient 1. Solution 12 log b m 32 log b 2n log b m 2 n log b m log b 2n 2 log b m 2 n 1 3 2 2 m 2 n log b m2n 3 1 2 2 2 n log b 3 m 2 1 3 2 2 n log b 3 log b 8n3 m m 1 2 3 Another Type of Problem • If loga3 = x and loga4 = y, express each log expression in terms of x and y. 1. loga12 • Loga(3•4) = loga3 + loga4 = x+y 2. Log34 • Log34 = loga4/loga3 = y/x Assignment • Pg. 211 – 213: #1 – 19 odd, 31, 33, 37 – 55 odd, 59 – 75 odd