1. Design: D Whitfield, www.pifactory.co.uk Logarithms – making complex calculations easy John Napier John Wallis Jost Burgi Johann Bernoulli

2. Design: D Whitfield, www.pifactory.co.uk 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 10 100 = 2 Number

3. Design: D Whitfield, www.pifactory.co.uk Logarithms 10 2 = 100 Base Logarithm Log 10 100 = 2 Number Logarithm Number Base y = b x Log b y = x 2 3 = 8Log 2 8 = 3 3 4 = 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

4. Design: D Whitfield, www.pifactory.co.uk 10 3 = 1000log 10 1000 = 3 2 4 = 16log 2 16 = 4 10 4 = 10,000log 10 10000 = 4 3 2 = 9log 3 9 = 2 4 2 = 16log 4 16 = 2 10 -2 = 0.01log 10 0.01 = -2 log 4 64 = 34 3 = 64 log 3 27 = 33 3 = 27 log 36 6 = 1 / 2 36 1/2 = 6 log 12 1= 012 0 = 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

5. Design: D Whitfield, www.pifactory.co.uk 10 3 = 1000log 10 1000 = 3 2 4 = 16log 2 16 = 4 10 4 = 10,000log 10 10000 = 4 3 2 = 9log 3 9 = 2 4 2 = 16log 4 16 = 2 10 -2 = 0.01log 10 0.01 = -2 log 4 64 = 34 3 = 64 log 3 27 = 33 3 = 27 log 36 6 = 1 / 2 36 1/2 = 6 log 12 1= 012 0 = 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

6. Design: D Whitfield, www.pifactory.co.uk 10 3 = 1000log 10 1000 = 3 2 4 = 16log 2 16 = 4 10 4 = 10,000log 10 10000 = 4 3 2 = 9log 3 9 = 2 4 2 = 16log 4 16 = 2 10 -2 = 0.01log 10 0.01 = -2 log 4 64 = 34 3 = 64 log 3 27 = 33 3 = 27 log 36 6 = 1 / 2 36 1/2 = 6 log 12 1= 012 0 = 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

7. Design: D Whitfield, www.pifactory.co.uk 10 3 = 1000log 10 1000 = 3 2 4 = 16log 2 16 = 4 10 4 = 10,000log 10 10000 = 4 3 2 = 9log 3 9 = 2 4 2 = 16log 4 16 = 2 10 -2 = 0.01log 10 0.01 = -2 log 4 64 = 34 3 = 64 log 3 27 = 33 3 = 27 log 36 6 = 1 / 2 36 1/2 = 6 log 12 1= 012 0 = 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

8. Design: D Whitfield, www.pifactory.co.uk 10 3 = 1000log 10 1000 = 3 2 4 = 16log 2 16 = 4 10 4 = 10,000log 10 10000 = 4 3 2 = 9log 3 9 = 2 4 2 = 16log 4 16 = 2 10 -2 = 0.01log 10 0.01 = -2 log 4 64 = 34 3 = 64 log 3 27 = 33 3 = 27 log 36 6 = 1 / 2 36 1/2 = 6 log 12 1= 012 0 = 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

9. Design: D Whitfield, www.pifactory.co.uk 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

10. Design: D Whitfield, www.pifactory.co.uk 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

11. Design: D Whitfield, www.pifactory.co.uk 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

12. Design: D Whitfield, www.pifactory.co.uk 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

13. Design: D Whitfield, www.pifactory.co.uk Change of base property Log a x = Log b x Log b a

14. Design: D Whitfield, www.pifactory.co.uk 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