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)

  aⁿ•a^m = a^(n+m)

  (aⁿ)^m = a^(n•m)

  a⁰ = 1, for a ≠ 0;

  a^(1/m) is the 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) = (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) = e^x 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 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) = 10^x is x = log(y)

  The inverse of the function y = f(x) = a^x is x = log_a(y)

  log_e(x) = log_e(10^{(log₁₀(x)}) = log₁₀(x)•log_e(10)

  log_a(x) = log_a(b^(log_b(x)) = log_b(x)•log_a(b)