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Properties of Logarithms Section 4.3

Properties of Logarithms Section 4.3. JMerrill, 2005 Revised, 2008. Rules of Logarithms If M and N are positive real numbers and b is ≠ 1:. The Product Rule : log b MN = log b M + log b N (The logarithm of a product is the sum of the logarithms)

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Properties of Logarithms Section 4.3

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  1. Properties of LogarithmsSection 4.3 JMerrill, 2005 Revised, 2008

  2. Rules of LogarithmsIf M and N are positive real numbers and b is ≠ 1: • The Product Rule: • logbMN = logbM + logbN (The logarithm of a product is the sum of the logarithms) • Example: log4(7 • 9) = log47 + log49 • Example: log (10x) = log10 + log x

  3. Rules of LogarithmsIf M and N are positive real numbers and b ≠ 1: • The Product Rule: • logbMN = logbM + logbN (The logarithm of a product is the sum of the logarithms) • Example: log4(7 • 9) = log47 + log49 • Example: log (10x) = log10 + log x • You do: log8(13 • 9) = • You do: log7(1000x) = log813 + log89 log71000 + log7x

  4. Rules of LogarithmsIf M and N are positive real numbers and b ≠ 1: • The Quotient Rule (The logarithm of a quotient is the difference of the logs) • Example:

  5. Rules of LogarithmsIf M and N are positive real numbers and b ≠ 1: • The Quotient Rule (The logarithm of a quotient is the difference of the logs) • Example: • You do:

  6. Rules of LogarithmsIf M and N are positive real numbers, b ≠ 1, and p is any real number: • The Power Rule: • logbMp = p logbM (The log of a number with an exponent is the product of the exponent and the log of that number) • Example: log x2 = 2 log x • Example: ln 74 = 4 ln 7 • You do: log359 = • Challenge: 9log35

  7. Prerequisite to Solving Equations with Logarithms • Simplifying • Expanding • Condensing

  8. Simplifying (using Properties) • log94 + log96 = log9(4 • 6) = log924 • log 146 = 6log 14 • You try: log1636 - log1612 = • You try: log316 + log24 = • You try: log 45 - 2 log 3 = log163 Impossible! log 5

  9. Using Properties to Expand Logarithmic Expressions • Expand: Use exponential notation Use the product rule Use the power rule

  10. Expanding

  11. Condensing • Condense: Product Rule Power Rule Quotient Rule

  12. Condensing • Condense:

  13. Bases • Everything we do is in Base 10. • We count by 10’s then start over. We change our numbering every 10 units. • In the past, other bases were used. • In base 5, for example, we count by 5’s and change our numbering every 5 units. • We don’t really use other bases anymore, but since logs are often written in other bases, we must change to base 10 in order to use our calculators.

  14. Change of Base • Examine the following problems: • log464 = x • we know that x = 3 because 43 = 64, and the base of this logarithm is 4 • log 100 = x • If no base is written, it is assumed to be base 10 • We know that x = 2 because 102 = 100 • But because calculators are written in base 10, we must change the base to base 10 in order to use them.

  15. Change of Base Formula • Example log58 = • This is also how you graph in another base. Enter y1=log(8)/log(5). Remember, you don’t have to enter the base when you’re in base 10!

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