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Properties of Logarithms

Properties of Logarithms. Book section 7.5. Properties of Logarithms. Product Property of Logarithms . Quotient Property of Logarithms Power Property of Logarithms Change of Base Formula Property of Equality for Logarithms. Examples: Product Property. log 2 (3x) log 5 ( xy ) ln (7ab).

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Properties of Logarithms

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  1. Properties of Logarithms Book section 7.5

  2. Properties of Logarithms • Product Property of Logarithms. • Quotient Property of Logarithms • Power Property of Logarithms • Change of Base Formula • Property of Equality for Logarithms

  3. Examples: Product Property • log2(3x) • log5(xy) • ln(7ab) log2(3) + log2(x) log5(x) + log5(y) ln(7) + ln(a) + ln(b)

  4. Examples: Quotient Property • log2(3/x) • log5(x/y) • ln(7a/b) • ln(7/ab) log2(3) - log2(x) log5(x) - log5(y) (ln(7) + ln(a)) - ln(b) ln(7) – (ln(a) + ln(b))

  5. Examples: Power Property • log2(3x2) • log5(x2y3) • ln(7a2b) log2(3) + 2log2(x) 2log5(x) + 3log5(y) ln(7) + 2ln(a) + ln(b)

  6. Putting it all together Remember: on top is added, on bottom is subtracted, and exponents become coefficients.

  7. Examples: Property of Equality • log(x + 3)= log(7 – x) • x + 3 = 7 – x • 2x = 4 • x = 2

  8. Examples: Property of Equality • log(3) + log(x – 2)= log(15) • log(3(x – 2))= log(15) • log(3x – 6)= log(15) • 3x – 6 = 15 • x = 7

  9. Examples: Change of Base

  10. Applications of Logarithmic Functions Unit 6.9

  11. Applications The magnitude of Stars The Richter Scale of Earthquake Intensity The intensity of sound

  12. Magnitude of Stars Over 2500 years ago(600BC), the astronomer Ptolemy grouped the visible stars into six categories according to their brightness. The first group contained stars that were brightest to the naked eye and the sixth group contained the faintest.

  13. Magnitude of Stars Over 2400 years later (in the mid-1800’s), astronomers formalized the classification as the brightest stars having a magnitude of 1 and the faintest group having a magnitude of 6. With the invention of the telescope they can now see stars that are fainter than before; thus, they extended the chart to include stars fainter than could be seen with the naked eye.

  14. Magnitude of Stars In measuring the brightness of stars at each magnitude level, they discovered that a star of magnitude 1 is 2.512 or 2.5121 times as bright as a star of magnitude 2. A star of magnitude 2 is 6.3 or 2.5122 times as bright as a star of magnitude 3. The model developed is m = 1+log2.512x

  15. Magnitude of Stars The model developed is m = 1+log2.512x How much brighter is the 1st magnitude star than a 6th magnitude star?

  16. Magnitude of Stars The model developed is m = 1+log2.512x How much brighter is the 1st magnitude star than a 6th magnitude star? A 1st magnitude star is about 100 times brighter than a 6th magnitude star

  17. Magnitude of Stars Using the magnitude model m = 1+log2.512x How much brighter is a star of magnitude 1 than a star of magnitude 9? A 1st magnitude star is about 1585.47 times as bright as a 9th magnitude star

  18. Magnitude of Stars Using the magnitude model m = 1+log2.512x What does it mean if a magnitude is negative? Since the brightest stars were given a magnitude of 1, a magnitude less than 1 must mean brighter than a first magnitude star. Many things are brighter than a first magnitude star, such as Venus, Jupiter, and the moon.

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