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Activity 38

Activity 38. Laws of Logarithms (Section 5.3, pp. 409-413). Laws of Logarithms:. Let a be a positive number, with a ≠ 1. Let A, B, and C be any real numbers with A > 0 and B > 0. Example 1:. Evaluate each expression:. Example 2:. Use the Laws of Logarithms to expand each expression:.

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Activity 38

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  1. Activity 38 Laws of Logarithms (Section 5.3, pp. 409-413)

  2. Laws of Logarithms: Let a be a positive number, with a ≠ 1. Let A, B, and C be any real numbers with A > 0 and B > 0.

  3. Example 1: Evaluate each expression:

  4. Example 2: Use the Laws of Logarithms to expand each expression:

  5. Example 3: Use the Laws of Logarithms to combine the expression into a single logarithm.

  6. Example 4: Use the Laws of Logarithms to combine the expression into a single logarithm.

  7. Example 5 (Forgetting): Ebbinghaus’s Law of Forgetting states that if a task is learned at a performance level P0, then after a time interval t the performance level P satisfies where c is a constant that depends on the type of task and t is measured in months.

  8. (a) Solve the equation for P.

  9. (b) Use Ebbinghaus’s Law of Forgetting to estimate a student’s score on a biology test two years after he got a score of 80 on a test covering the same material. Assume c = 0.3.

  10. Example 6 (Biodiversity): Some biologists model the number of species S in a fixed area A (such as an island) by the Species-Area relationship where c and k are positive constants that depend on the type of species and habitat. (a) Solve the equation for S.

  11. (b) Use part (a) to show that if k = 3 then doubling the area increases the number of species eightfold. Let the area be A0 Then double the area is 2A0 Number of species when the Area is A0 is Number of species when the Area is 2A0

  12. Change of Base: For some purposes, we find it useful to change from logarithms in one base to logarithms in another base. One can prove that:

  13. Example 7: Use the Change of Base Formula and common or natural logarithms to evaluate each logarithm, correct up to five decimal places:

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