Design: D Whitfield, Logarithms – making complex calculations easy John Napier John Wallis Jost Burgi Johann Bernoulli

Design: D Whitfield, Logarithms 10 2 = 100 “10 raised to the power 2 gives 100” Base Index Power Exponent Logarithm “The power to which the base 10 must be raised to give 100 is 2” “The logarithm to the base 10 of 100 is 2” Log = 2 Number

Design: D Whitfield, Logarithms 10 2 = 100 Base Logarithm Log = 2 Number Logarithm Number Base y = b x Log b y = x 2 3 = 8Log 2 8 = = 81Log 3 81 = 4 Log 5 25 =25 2 = 25 Log 9 3 = 1 / 2 9 1/2 = 3 log b y = x is the inverse of y = b x

Design: D Whitfield, = 1000log = = 16log 2 16 = = 10,000log = = 9log 3 9 = = 16log 4 16 = = 0.01log = -2 log 4 64 = 34 3 = 64 log 3 27 = 33 3 = 27 log 36 6 = 1 / /2 = 6 log 12 1= = 1 p = q 2 log q p = 2 x y = 2log x 2 = y p q = rlog p r = q log x y = zx z = y log a 5 = ba b = 5 log p q = rp r = q c = log a bb = a c

Design: D Whitfield, = 1000log = = 16log 2 16 = = 10,000log = = 9log 3 9 = = 16log 4 16 = = 0.01log = -2 log 4 64 = 34 3 = 64 log 3 27 = 33 3 = 27 log 36 6 = 1 / /2 = 6 log 12 1= = 1 p = q 2 log q p = 2 x y = 2log x 2 = y p q = rlog p r = q log x y = zx z = y log a 5 = ba b = 5 log p q = rp r = q c = log a bb = a c

Design: D Whitfield, = 1000log = = 16log 2 16 = = 10,000log = = 9log 3 9 = = 16log 4 16 = = 0.01log = -2 log 4 64 = 34 3 = 64 log 3 27 = 33 3 = 27 log 36 6 = 1 / /2 = 6 log 12 1= = 1 p = q 2 log q p = 2 x y = 2log x 2 = y p q = rlog p r = q log x y = zx z = y log a 5 = ba b = 5 log p q = rp r = q c = log a bb = a c

Design: D Whitfield, = 1000log = = 16log 2 16 = = 10,000log = = 9log 3 9 = = 16log 4 16 = = 0.01log = -2 log 4 64 = 34 3 = 64 log 3 27 = 33 3 = 27 log 36 6 = 1 / /2 = 6 log 12 1= = 1 p = q 2 log q p = 2 x y = 2log x 2 = y p q = rlog p r = q log x y = zx z = y log a 5 = ba b = 5 log p q = rp r = q c = log a bb = a c

Design: D Whitfield, = 1000log = = 16log 2 16 = = 10,000log = = 9log 3 9 = = 16log 4 16 = = 0.01log = -2 log 4 64 = 34 3 = 64 log 3 27 = 33 3 = 27 log 36 6 = 1 / /2 = 6 log 12 1= = 1 p = q 2 log q p = 2 x y = 2log x 2 = y p q = rlog p r = q log x y = zx z = y log a 5 = ba b = 5 log p q = rp r = q c = log a bb = a c

Design: D Whitfield, Laws of logarithms Every number can be expressed in exponential form… every number can be expressed as a log Let p = log a x and q = log a y So x = a p and y = a q xy = a p+q p + q = log a (xy) p + q = log a x + log a y = log a (xy) log a (xy) = log a x + log a y

Design: D Whitfield, Laws of logarithms Every number can be expressed in exponential form… every number can be expressed as a log Let p = log a x and q = log a y So x = a p and y = a q xy = a p-q p - q = log a ( x / y ) p - q = log a x - log a y = log a ( x / y ) log a ( x / y ) = log a x - log a y

Design: D Whitfield, Laws of logarithms Every number can be expressed in exponential form… every number can be expressed as a log Let p = log a x and q = log a x So x = a p and x = a q x 2 = a p+q p + q = log a (x 2 ) p + q = log a x + log a x = log a (x 2 ) log a x n = nlog a x

Design: D Whitfield, Laws of logarithms Every number can be expressed in exponential form… every number can be expressed as a log log a ( x / y ) = log a x - log a y log a (xy) = log a x + log a y log a x n = nlog a x a m.a n = a m+n a m /a n = a m-n (a m ) n = a m.n

Design: D Whitfield, Change of base property Log a x = Log b x Log b a

Design: D Whitfield, Solving equations of the form a x = b 3 x = 9 4 x = 64 5 x = 67 Solve by taking logs: log5 x = log67 xlog5 = log67