# Useful Formulae

ap ap a pq aq a Given some value a, we can define a squared and a cubedas follows a2 axa, a3 axaxa. In the expression an,nis the exponent. Here are the basic exponential formul a0 1 apaq apq abp apbp a 1 n na apq apq amn nam The basea logarithmof x, loga x y is the quantity which satisfies ay x. Because a0 1 and a1 a,we always have logaa 1and loga1 0. Here are the essential logarithmic formul loga xy loga x loga y loga loga x loga y loga xk kloga x loga loga x loga loga x logka logma logkm Any positive base except 1 can be used but most common are 10 and the constant e 2.718...,which occurs widely throughout the natural world,often in processes of growth and decay. loge is usually just written logorln. Logarithms allow multiplication and division of numbers by the addition or subtraction of exponents. 18 EXPONENTIALSLOGARITHMS growth and decay n x 1 ap n1 y x x1 a b n n b n 19