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LOGARITHMS. Another way to play with EXPONENTS. DEFINITION. What is it…. A LOGARITHM IS AN EXPONENT. If x = b y then log b (x) = y. PROPERTIES. What makes a logarithm tick!. MULTIPLICATON ADDITION.

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logarithms

LOGARITHMS

Another way to play with EXPONENTS

definition
DEFINITION

What is it…

a logarithm is an exponent
A LOGARITHM IS AN EXPONENT
  • If x = by then logb(x) = y
properties
PROPERTIES

What makes a logarithm tick!

multiplicaton addition
MULTIPLICATON ADDITION
  • A major property of logarithms is that they map multiplication to addition, as a result of the identity

bxx by = b(x + y)

  • which by taking logarithms becomes

logb(bx x by) = logb(b(x + y)) = x + y = logb(bx) + logb(by)

taking the log of an exponent
TAKING THE LOG OF AN EXPONENT
  • If you take the log of a number with an exponent, the exponent becomes a coefficient!
  • logb(cp) = p logb(c)
natural logarithms and common logarithms
NATURAL LOGARITHMS and COMMON LOGARITHMS
  • A logarithm can have any base
    • We will concentrate on 2
  • Common Logs have a base of 10
    • log104 is written as log 4
  • Natural Logs have a base of e
    • e = 2.7182818…
    • Also called Euler’s Number after Leonhard Euler
    • loge4 is written as ln 4
even more properties
EVEN MORE PROPERTIES

More exponential behavior…

products
PRODUCTS
  • Multiplying becomes addition
    • logbmn = logbm + logbn
    • For example: log (3· 4) = log 3 + log 4
quotients
QUOTIENTS
  • Division becomes subtraction
    • logb = logbm - logbn
    • Example: log = log 4 – log3
try this
TRY THIS…
  • log4(64y6)1/3 = ?
final tricks
FINAL TRICKS

A few more things to know…

how to calculate
HOW TO CALCULATE
  • Your calculator has a button for common and natural logs.
  • Other logs can use the following property:
    • log6 x = (log x)/(log 6)
transformations translation
TRANSFORMATIONS-TRANSLATION
  • Compare y = log x to y = log (x + 2)
  • Compare y = log x to y = (log x) + 2
transformations dilation
TRANSFORMATIONS-DILATION
  • Compare y = log x to y = 2 log x