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Warm-Up

Warm-Up. 1) Write in exponential form. 2) Write in logarithmic form. log 27 9 = x. 5 x = 2003. 3) Evaluate. 4) Write the Equation that models this situation:

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Warm-Up

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  1. Warm-Up 1) Write in exponential form. 2) Write in logarithmic form. log279 = x 5x = 2003 3) Evaluate. 4) Write the Equation that models this situation: The current world population of Oompa-loompas is 7,986. If the Ooompa-loompa population grows steadily at a rate of 19% per year, what will their total population be in 10 years?

  2. 8.5 Properties of Logarithms

  3. Properties of Logarithms loga1 = 0 because a0 = 1 logaa = 1 because a1 = a logaax = x If loga x= loga y then x =y Product Property Quotient Property Power Property Change-of-Base

  4. Remember Common Logarithm?? log10 Is the “common” logarithm This is the LOG button on the calculator Sometimes we’re lazy and we don’t even write the base… log107 = log 7

  5. Basic Properties of Logarithms 1) loga1 = 0 because a0 = 1 2) logaa = 1 because a1 = a

  6. 3) One-to-one Property: If loga x = loga y then x=y

  7. Product Property Express as a sum of logarithms. 1) loga MN = loga M + loga N 2) logb AT = logb A + logb T 3) log MATH = log M + log A + log T + log H

  8. Ex. Express as a single logarithm 4) log5 19 + log5 3 = log5 (19*3) 5) log C + log A + log B + log I + log N = log CABIN

  9. Ex. Express as a sum of logarithms, then simplify 6) log2 (4*16) = log2 4 + log216 = 2 + 4 = 6

  10. Ex. 7 Use log53 = 0.683 and log57 = 1.209 to approximate… log5 (21) = log5 (3*7) = log5 3 + log5 7 = 0.683 + 1.209 = 1.892

  11. Quotient Property

  12. 1) Ex. Express as the difference of logs 2)

  13. Ex. 3 Use log53 = 0.683 and log57 = 1.209 to approximate… = log5 3 - log5 7 = 0.683 - 1.209 = -0.526

  14. Power Property

  15. try these: conclusions...

  16. Ex. Express as a product. = -5 * logb9 4) 5)

  17. Ex. 6 Use log53 = 0.683 and log57 = 1.209 to approximate… log5 49 = log5 72 = 2 log5 7 = 2(1.209) = 2.418

  18. Ex. 7 Expand. log105x3y log105 + log10x3 + log10y log105 + 3 log10x + log10y

  19. Ex. 8 Expand • Simplify the division. • Simplify the multiplication of 4 • Change the radical sign to an exponent • Express the exponent as a product

  20. Ex. Condense. 9) 10)

  21. Ex 11 Condense • Express all products as exponents • Change the fractional exponent to a radical sign. • Simplify the subtraction. • Simplify the addition.

  22. Warning!!! Be careful!!!

  23. Change-of-Base this allows us to change from any base to base 10

  24. Ex. 1 Evaluate. Round to four decimal places. = 1.7712 Be careful about the parenthesis in the calculator… log(7) / log(3)

  25. Ex. 2 Evaluate. Round to four decimal places. = 0.2626

  26. Assignment:page 496#14-20 even, #30-70x4

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