+124 Given log53 = 0.6826 and log58=1.2920, evaluate the expressions.
(a) log52
(b) log5(75/8)
Thank you so much for checking this question out, can you please help me solve it
+177 Simply use some creative algebraic manipulation with the property of logarithms to find an approximation for the other expressions.
a) \(\log_{5}2\)
There are probably many approaches that would lead to the same answer, but here is how I would approach this one:
\(\log_{5}8 =\log_{5}2^3 \\ \log_58 = 3\log_52 \\ 1.2920 = 3\log_52 \\ \log_52 = \frac{1.2920}{3} \approx 0.4307\)
b) \(\log_5 \frac{75}{8}\)
Luckily, we use the result from before to help this particular problem.
\(\log_5 \frac{75}{8} = \log_5 75-\log_5 8 \\ \log_5 \frac{75}{8} = \log_5 (3*25) - \log_5 8\\ \log_5 \frac{75}{8} = \log_5 3 + \log_5 25 - \log_5 8\\ \log_5 \frac{75}{8} = 0.6826 + 2 - 1.2920 \approx 1.3906 \\ \)
You can check to determine if your answer is reasonable by using a calculator and evaluating \(\log_5 2\) and \(\log_5 \frac{75}{8}\). The decimal expansion should be match up for the first few digits.
+177 Simply use some creative algebraic manipulation with the property of logarithms to find an approximation for the other expressions.
a) \(\log_{5}2\)
There are probably many approaches that would lead to the same answer, but here is how I would approach this one:
\(\log_{5}8 =\log_{5}2^3 \\ \log_58 = 3\log_52 \\ 1.2920 = 3\log_52 \\ \log_52 = \frac{1.2920}{3} \approx 0.4307\)
b) \(\log_5 \frac{75}{8}\)
Luckily, we use the result from before to help this particular problem.
\(\log_5 \frac{75}{8} = \log_5 75-\log_5 8 \\ \log_5 \frac{75}{8} = \log_5 (3*25) - \log_5 8\\ \log_5 \frac{75}{8} = \log_5 3 + \log_5 25 - \log_5 8\\ \log_5 \frac{75}{8} = 0.6826 + 2 - 1.2920 \approx 1.3906 \\ \)
You can check to determine if your answer is reasonable by using a calculator and evaluating \(\log_5 2\) and \(\log_5 \frac{75}{8}\). The decimal expansion should be match up for the first few digits.
