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Properties Continued

Properties Continued. 5 – Product Law: log c mn = log c m + log c n. Simplify: log 3 10 + log 3 2. log 3 (10 . 2). log 3 (20). log20 log3. = 2.73. Example. Solve for x: log 2 2 + log 2 5 = x log 2 (2x5) = x log 2 10 = x log10 = x log2 3.32 = x.

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Properties Continued

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  1. Properties Continued

  2. 5 – Product Law:logcmn = logcm + logcn Simplify:log310 + log32 log3(10.2) log3(20) log20log3 = 2.73

  3. Example Solve for x: log22 + log25 = x log2(2x5) = x log210 = x log10 = x log2 3.32 = x

  4. 6 – Quotient Law:logc(m/n) = logcm - logcn Simplify:log25 – log23 log2(5/3) log(5/3)log2 = 0.74

  5. Example Solve for x: log333 - log311 = x log3(33/11) = x log33 = x x = 1

  6. Example Solve for x: log296 - log23 = x log2(96/3) = x log232 = x 2x = 32 x = 5

  7. 7 –Law of Powers:logcmn = nlogcm Simplify:xlog28x x2log2(8) x2log(8) log2 = 3x2

  8. Example Simplify: log243 3log24 3(2)(since log24=2) 6

  9. Example Simplify: log23√7 = log27⅓ = ⅓log27 = ⅓(2.81) = 0.94

  10. Example log7493 = 3log749 = 3(2)(since log749=2) = 6

  11. 8 – Change of Base Law:logcx = logx(we already know this one!)logc

  12. Example 7x = 400 log7400 = x x = log400log7 x = 3.07

  13. Examples: Simplify each expression into a single logarithm: a) log4 – 2log8 b) 5lny -3lnx c) log2x + (logy – log4) d) 2(log2x – logy) – (log3 + 2log5)e) 2log3 + log2 – log6 – log4 f) 3lny -2lny2 + lny3

  14. a) log4 – 2log8 b) 5lny -3lnx c) log2x + (logy – log4) d) 2(log2x – logy) – (log3 + 2log5)

  15. e) 2log3 + log2 – log6 – log4 f) 3lny -2lny2 + lny3

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