## Exponentials and Logarithms

• Powers

an, for n = 1, 2, 3, ..., is a short way of saying a•a•a•...•a, where a is multiplied by itself n times

a-n = 1/(an)

anam = a(n+m)

(an)m = a(n•m)

a0 = 1, for a ≠ 0;

a(1/m) is the mth 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 ax, a ≥ 0, where x is irrational, by using rational approximations n/m to x and computing a(n/m) = (an)(1/m), taking the positive root.

• Compound Interest Problems

A = P•( 1 + r )n, where 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 )(n•4) for quarterly interest where n is the number of years

A = P•( 1 + r/12 )(n•12) for monthly interest where n is the number of years

A = P•( 1 + r/m )(n•m) for m compoundings per year where n is the number of years

The 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) = ex maps the real numbers into the non-negative real numbers. The inverse of this function is the natural logarithm x = ln(y) which maps the non-negative real numbers into the real numbers.

Another name for ln is loge

ln(1) = 0

ln(e) = 1

ln(ab) = bln(a)

ln(ab) = ln(a) + ln(b)

• The log with other bases

The inverse of the function y = f(x) = 10x is x = log(y)

The inverse of the function y = f(x) = ax is x = loga(y)

loge(x) = loge(10(log10(x)) = log10(x)•loge(10)

loga(x) = loga(b(logb(x)) = logb(x)•loga(b)