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Understanding Logarithm Properties and Simplification Techniques

This section covers fundamental properties of logarithms, including the product, quotient, and power rules. Key examples illustrate how to simplify logarithmic expressions using these properties. It discusses rewriting logarithms in terms of different bases and gives examples on expanding and condensing logarithmic expressions. Additionally, the change-of-base formula is introduced for evaluating logs with different bases. This resource aims to enhance comprehension and application of logarithmic rules in various mathematical contexts.

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Understanding Logarithm Properties and Simplification Techniques

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  1. Section 8.5 Properties of Logarithms

  2. logb(MN)= logbM + logbN Ex: log4(15)= log45 + log43 logb(M/N)= logbM – logbN Ex: log3(50/2)= log350 – log32 logbMr = r logbM Ex: log7 103 = 3 log7 10 logb(1/M) =logbM-1= –1 logbM= – logbM Ex: log11 (1/8) = log11 8-1 = – 1 log11 8 = – log11 8 Properties of Logarithms

  3. Examples:Rewrite in terms of log57 and log53

  4. Examples of Logarithms • Simplify log 7 + log 4 – log 2 = log 7*4 = log 14 2 • Simplify ln e2 = 2 ln e = 2 logee = 2 * 1 = 2 • Simplify e 4 ln 3 - 3 ln 4= e ln 34 - ln 43 = e ln 81/64 = e loge81/64 = 81/64

  5. Expanding Logarithms Expand :

  6. Condensing Logarithms Condense: log 6 + 2 log 2 – log 3 log 6 + 2 log 2 – log 3 = log 6 + log 22 – log 3

  7. logam logbm = -------- logab log712 = log 12 log 7 OR Change-of-Base Formula • log712= ln 12 • ln 7

  8. Example • Evaluate: 1) log37 = 2) log410 =

  9. Assignment Section 8.5: page 496 – 497 # 15 – 72 (every 3rd)

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