~LAWS OF EXPONENTS~

(a^x)(a^y)= a^(x+y), a^x/a^y=a^(x-y), (a^x)^y=a^(xy)
(ab)ⁿ=(aⁿ)(bⁿ), (a/b)ⁿ= aⁿ/bⁿ, 1/aⁿ=a^(-n)

a = b if and only if aⁿ=bⁿ raising each side of an equation
to the same power---mainly used to eliminate radicals
(fractional exponents)

LOGARITHMIC PROPERTIES (properties of logs)

log_a(xy)=log_ax + log_ay
(the log of a product is = the sum of the logs of each factor)
(or, multiplication in the domain is equivalent to addition in the range)

  log_a(x/y)=log_ax - log_ay
  (the log of a quotient is = the difference of the logs in that order)
  (or, division in the domain is equivalent to subtraction in the range)
  (don't confuse log_ax / log_ay with this one)
  (this is the quotient of two logs) (the division of two range values)

  log_a(uⁿ)=nlog_au
  (very useful for "knocking powers down")

  A = B if and only if log_aA=log_aB
  (taking the log of both sides of an equation)

  *ln u = log_eu (natural log)
  (special symbol for the natural log) (on calculator)

   log u = log₁₀u (common log)
   (base is not written) (on calculator)

   *ln e = 1 (important feature of the number e)

   *ln A = B if and only if A = e^B (used much---very popular)
    (both ways)

    *ln(e^u)=u

    *log_aa=1

    *u= B^[log_Bu] (B raised to the power of log_Bu equals u)

    *log_AB= (ln B) / (ln A) (converting any log to natural logs)
     (could convert to any other base log by replacing ln by that base log)
     (i.e., log_AB=[log₇B] / [log₇A])

    ~Note: many students "mess up" calculus problems because of the
                misuse of properties of expos & logs...so, know them well...

~Note: Many times, these properties are used in reverse (i.e., instead of "knocking down" a power using a log property, we place the power "back up" on the quantity). So, know how to use them in both directions.