Logarithms

Introduction

The logarithm is the power to which a base must be raised to obtain a particular number. That is:

y=bt iff t=logby

Manipulating Logarithms

The following rules/results make it relatively easy to manipulate logarithms:

  1. loguv=logu+logv
  2. logu/v=logulogv
  3. loguk=klogu for u>0
  4. logbu=(logba)(logau) for u>0
    To prove this result let u=ap (so that p=logau). Then:
    logbu=logbap=plogba=(logba)(logau)
  5. logbu=1logub
  6. Combining the previous two rules yields the following “change of base rule”:
    logbu=logaulogab for u>0
  7. ddulogbu=1(ulnb) where ln denotes the natural logarithm (i.e., loge).