- Powers
*a*, for^{n}*n*= 1, 2, 3, ..., is a short way of saying*a•a•a•...•a*, where a is multiplied by itself*n*times*a*= 1/(^{-n}*a*)^{n}*a*•^{n}*a*=^{m}*a*^{(n+m)}(

*a*)^{n}=^{m}*a*^{(n•m)}*a*= 1, for a ≠ 0;^{0}*a*is the^{(1/m)}*m*^{th}root of*a*, a number which when multiplied by itself*m*times will result in*a*. If*m*is even and*a*> 0; there are two roots.We can approximate

*a*,^{x}*a*≥ 0, where*x*is irrational, by using rational approximations*n/m*to*x*and computing*a*= (^{(n/m)}*a*)^{n}, taking the positive root.^{(1/m)} - Compound Interest Problems
*A*=*P*•( 1 +*r*), where^{n}*P*is the original principal amount,*r*is the interest rate per interest period, and*n*is the number of interest periods.*A*is the resulting total amount.*A*=*P*•( 1 +*r/4*)for quarterly interest where^{(n•4)}*n*is the number of years*A*=*P*•( 1 +*r/12*)for monthly interest where^{(n•12)}*n*is the number of years*A*=*P*•( 1 +*r/m*)for^{(n•m)}*m*compoundings per year where*n*is the number of yearsThe number given by the limit as

*m*goes to infinity of (1 + 1/*m*)^{m}is called*e*, approximately 2.718281828 - The natural logarithm
The function

*y*= f(*x*) =*e*maps the real numbers into the non-negative real numbers. The inverse of this function is the natural logarithm^{x}*x*=*ln*(*y*) which maps the non-negative real numbers into the real numbers.Another name for

*ln*is*log*_{e}*ln*(1) = 0*ln*(*e*) = 1*ln*(*a*) =^{b}*b*•*ln*(*a*)*ln*(*a*•*b*) =*ln*(*a*) +*ln*(*b*) - The log with other bases
The inverse of the function

*y*= f(*x*) = 10is^{x}*x*=*log*(*y*)The inverse of the function

*y*= f(*x*) =*a*is^{x}*x*=*log*(_{a}*y*)*log*(_{e}*x*) =*log*(10_{e}^{(log10(x)}) =*log*_{10}(*x*)•log_{e}(10)*log*(_{a}*x*) =*log*(_{a}*b*^{(logb(x)}) =*log*_{b}(*x*)•log_{a}(*b*)