Math  /  Algebra

QuestionProblems 17 - 22, Find the exact value of the logarithm without using a calculator. If this is not possible, state the reason.
17. log5125log55\log _{5} 125-\log _{5} 5
18. 6lne5+4lne26 \ln e^{5}+4 \ln e^{-2}
19. ln1e\ln \frac{1}{\sqrt{e}}
20. log3(27)\log _{3}(-27)
21. log2321\log _{2} \sqrt[1]{32}
22. log3(181)\log _{3}\left(\frac{1}{81}\right)

Problems 23 - 24, Solve.
23. Mrs. Adams gave her students a quiz on logarithms. Every week for three months they took another quiz to see how much they remembered. The average scores of the students can be modeled by the human memory model S(t)=8712log(t+1)S(t)=87-12 \log (t+1) for 0t120 \leq t \leq 12 where tt is the time in weeks. A. Find the average score on the original quiz. s(0)=8712log(0+1)87s(0)=87-12 \log (0+1)^{87} B. What was the average score after 1 month ( 4 weeks)? S(4)=8712log(4+1)=878.39=78.61S(4)=87-12 \log (4+1)=87-8.39=78.61 C. Find the average score at the end of the 12 weeks. S(12)=812log(12+1)=8713.37=73.63S(12)=8-12 \log (12+1)=87-13.37=73.63
24. The Richter scale model for measuring magnitude RR of an earthquake is modeled by the equation R=log(ar)+BR=\log \left(\frac{a}{r}\right)+B, where aa is the amplitude in micrometers, TT is the period in seconds, and BB represents the dampening effect (weakening) of the wave due to the distance from the epicenter of the quake. A. Find the magnitude RR of a quake where a=325,T=4a=325, T=4 and B=3.25B=3.25 B. Find the magnitude RR of a quake where a=230,T=2a=230, T=2 and B=4.5B=4.5

Studdy Solution
Solve problem 24B: Find the magnitude RR of a quake where a=230,T=2a=230, T=2 and B=4.5B=4.5.
- Substitute the values into the Richter scale model R=log(aT)+BR = \log\left(\frac{a}{T}\right) + B.
R=log(2302)+4.5 R = \log\left(\frac{230}{2}\right) + 4.5
- Calculate 2302=115\frac{230}{2} = 115.
- Approximating log1152.06070\log 115 \approx 2.06070.
R2.06070+4.5=6.56070 R \approx 2.06070 + 4.5 = 6.56070

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