Understanding Laws of Logarithms: Exponents vs. Logarithms
Laws of Logarithms explain the properties and rules governing logarithmic functions, involving evaluations, conversions, additions, subtractions, and comparisons with exponent laws. Through examples, the laws of logarithms are applied to simplify expressions and evaluate logarithmic equations. The relationship between exponents and logarithms is explored, showcasing how these concepts are interconnected.
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8.3 Laws of Logarithms For x > 0 and b 1, Evaluation Logarithms Common Log logb1 = 0 logbb = 1 logbbm= m b logbx= x logb0 is not defined logb(-x) is not defined Examples log 1 = 0 log 10 = 1 log 10s= s 10 log x= x log 4 4 = 1 log 8 1 = 0 3 log 3 6 = 6 log 5 5 3 = 3 2 log 2 7 = 7 Logbx Logx Logax = Log2x = Logba Log2 Math 30-1 1
Addition Law of Logarithms loga(xy) = logax + logay Convert x = apand y = aq xy = ap+q Convert p + q = loga(xy) p + q = logax + logay Let p = logax and q = logay loga(xy) = logax + logay log7x = log7+ logx = log2 log100 + log200 Math 30-1 3
Express as the sum of two logarithms. 3 log 42 log 2 log 21 + log 3 log 14 + log 6 log 7 + 3 3 3 3 3 3 Express as the sum of two logarithms. 5 log 20 log 2 log 10 + log 4 log 5 + 5 5 5 5 ( ( ) ) ( ) 8 ( ) 8 ( ) 4 ( ) 4 8 4 log log log 2 2 2 8 4 = + log log log 2 2 2 Math 30-1 4
Subtraction Law of Logarithms loga(x/y) = logax - logay Convert x = apand y = aq x/y = ap-q Convert p - q = loga(x/y) p - q = logax - logay Let p = logax and q = logay loga(x/y) = logax - logay = log3x log32 x 2 log3 Math 30-1 6
Express as the difference of two logarithms. 3 log 4 log 8 log 2 log 12 log 3 3 3 3 3 Express as the difference of two logarithms. 5 log 2 log 10 log 5 log 20 log 10 5 5 5 5 8 4 8 2 ( ) 8 ( ) 8 ( ) 4 ( ) 4 log log log 2 2 2 ( ) = log log log 2 2 2 Math 30-1 7
Comparing Exponent Laws to Laws of Logarithms loga(xy) = logax + logay am.an = am+n loga(x/y) = logax - logay am/an = am-n 27 3 9 as a sum and difference of logarithms Express log3 27 3 9 log3 = log327 + log33 - log39 = 3 + 1 - 2 = 2 Math 30-1 8
Applying Laws of Logarithms Express as a single log: A log3A - log3B - log3C = = log3 BC AC B log3A - log3B + log3C = = log3 Evaluate: log210 + log212.8 = log2(10 x 12.8) = log2(128) = log2(27) = 7 Simplify: log550 - log510 50 10 log 5 log55 = 1 Math 30-1 9
Simplifying Logarithms Simplify: log45a + log48a3 - log410a4 5 log 10 3 8 a a a log44 4 40 10 a a log 4 4 4 4 = 1 Given log79 = a, determine an expression in terms of a for log763. log763 = log7(9 x 7) = log79 + log77 = a + 1 If and , express each of the following in terms of x and y = = log 6 log 4 x y a a log 384 a log 9 a Math 30-1 10
Math 30-1 11
Power Law: logb(xn) = nlogbx = logbx 1 logbx 1 x logbmn = n logbm logb n d= =n logbm dlogbm logbm3= logb(m m m) = logbm+ logbm+ logbm = 3logbm 1/3 log24 logbm3= 3logbm logbma 3log25 ( ) ( )( ) 2 = Math 30-1 log log log m m m b b b 12
Applying the Power Laws Given that log3a = 6 and log3b = 5 determine the value of log3(9ab2), where a, b > 0 ( 3 log 9ab 3 3 log 9 log log a ( ) 3 3 log 3 log a = + ( ) 2 6 2 5 = + + = ) = + + 2 b 2 3 2log + 2 b 3 18 Write as a single logarithm, where x, y, z > 0 log 2log 3log 2 1 2 z = + + 2 3 log log log x z y x y 2 3 2 3 x y x y = log or log 1 2 z z Math 30-1 13
4 log6 20 Given log62 = a and log65 = b rewrite in terms of a and b. = 1 4log62 2 5 = 1 4log62+ log62 +log65 =1 4 ( ) 4 log6 20 1 4 log 20 6 a + a +b ( ) =2a +b 4 Math 30-1 14
The expression is equivalent to ( 3 )( ) log log x x 3 ( )2 ( )2 2 2 log log x x log log x x 3 9 3 9 Evaluate: 2 1 8 ( ( ) ) ( ( ) ) ) ) 3 a) 3log2 16 log228+ + log2 b) 4 3 = = 3log22 = = 3 4 2 = = log228+ +log22 3 = = log228+ +log22 6 = = log228 2 6 ( ( = = log222 = 2 3 = 4 Math 30-1 15
Assignment: State whether the following are True or False for logarithms to every base. a) log 2 + log 3 = log 5 False b) log 4 + log 3 = log 12 True c) log 10 + log 10 = log 100 True d) log 2 x log 3 = log 6 False e) log 32 + log 3-2 = 0 True log5 3= =log5 log3 log8 log2= = log4 f) False g) False, think of change of base. Math 30-1 16
Assignment State whether the following are True or False for logarithms to every base. a) log 5-2 = -2log 5 True Page 400 1a,c, 2, 3, 5, 6, 8, 9, 10, 11, 12, log4 = =2 3log8 b) True 1 3log11 = =log11 log5 = =1 2log10 log1 5 log5 = = log25 c) False 3 d) False e) True log10 ( ) 2= 2log10 f ) False Math 30-1 17
