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LOGARITHMS

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

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  1. LOGARITHMS Another way to play with EXPONENTS

  2. DEFINITION What is it…

  3. A LOGARITHM IS AN EXPONENT • If x = by then logb(x) = y

  4. PROPERTIES What makes a logarithm tick!

  5. 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)

  6. 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)

  7. 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

  8. EVEN MORE PROPERTIES More exponential behavior…

  9. PRODUCTS • Multiplying becomes addition • logbmn = logbm + logbn • For example: log (3· 4) = log 3 + log 4

  10. QUOTIENTS • Division becomes subtraction • logb = logbm - logbn • Example: log = log 4 – log3

  11. TRY THIS… • log4(64y6)1/3 = ?

  12. FINAL TRICKS A few more things to know…

  13. 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)

  14. GRAPHING

  15. TRANSFORMATIONS-TRANSLATION • Compare y = log x to y = log (x + 2) • Compare y = log x to y = (log x) + 2

  16. TRANSFORMATIONS-DILATION • Compare y = log x to y = 2 log x

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