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12.3 Properties of Logarithms

12.3 Properties of Logarithms. Algebra 2 Mrs. Spitz Spring 2007. Objectives:. Solve equations or simplify and evaluate expressions using properties of logarithms. Assignment. Worksheet 12.3. Application.

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12.3 Properties of Logarithms

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  1. 12.3 Properties of Logarithms Algebra 2 Mrs. Spitz Spring 2007

  2. Objectives: • Solve equations or simplify and evaluate expressions using properties of logarithms.

  3. Assignment • Worksheet 12.3

  4. Application • Acid rain is a very serious problem in many parts of the world. It raises the pH level of lakes and streams, killing fish and other waterlife. The pH scale measures acidity and is a logarithmic scale. Lower pH numbers represent acids, and higher pH levels represent bases. Acid rain has a pH of about 4.2, while neutral water has a pH of 7. So, acid rain is about 630 times more acidic than neutral water, since 107-4.2 630. The pH scale is an example of logarithmic scale.

  5. FYI • A large portion of acid rain is the result of the burning of fossil fuels. Finding other sources of energy will help solve the problem. • Since logarithms are exponents, the properties of logarithms can be derived from the properties of exponents that you already know.

  6. Product of Powers • The product of powers is found by adding exponents. So it makes sense that the logarithm of a product is found by adding the logarithms. So, log 3 (9 27) = log 3 9 +log 327)

  7. Product Property of Logarithms • For all positive numbers m, n and b, where b ≠ 1, log b mn = log b m + log b n • To prove this property, let bx = m and by = n. Then log b m = x and log b n = y Property of exponents Prop. =/log functions Def. inverse function Substitution

  8. Ex. 1a: Given log 2 5 = 2.322, find each logarithm.

  9. Ex. 1b: Given log 2 5 = 2.322, find each logarithm.

  10. Quotient of powers • To find the quotient of powers, you subtract the exponents. So, to find the logarithms of a quotient, you subtract the logarithms.

  11. Quotient Property of Logarithms • So, the last example illustrated the quotient property of Logarithms • For all positive numbers m, n, and b, where b ≠ 1,

  12. To prove this property . . .

  13. The pH of a solution is related to the number of grams atoms of hydrogen ions, H+, by the formula . If the pH level of a lake is 5, how much more acidic is it than neutral water, that has a pH of 7? The pH of neutral water is 7. Since , there are 10-7 gram atoms of hydrogen ions in a liter of neutral water. Similarly, since , there are 10-5 gram atoms of hydrogen ions in a liter of lake water. Ex. 2: Application/Chemistry

  14. Continued • If we divide these two numbers, we will find out how much stronger the lake water is than neutral water. So the lake water is 100 times more acidic than the neutral water.

  15. Ex. 3: Given log 12 9 = 0.884 and log12 18 = 1.163, find each logarithm.

  16. The quotient property can be used to solve equations involving logarithms. Ex. 4: Solve log 5 4 + log5 x = log5 36

  17. Power of a Power • The power of a power is found by multiplying the two exponents. This suggests that the logarithm of a power is found by multiplying the logarithm and the exponent. OR

  18. Power Property of Logarithms • For any real number p and positive numbers m and b where b ≠ 1, log b mp = p  log b m

  19. Ex. 5: Solve

  20. Ex. 6: Solve

  21. Ex. 6: CONTINUED Both of these logs are undefined, so -4 is NOT A SOLUTION!

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