Given log_{5}3 = 0.6826 and log_{5}8=1.2920, evaluate the expressions.

(a) log_{5}2

(b) log_{5}(75/8)

Thank you so much for checking this question out, can you please help me solve it

thebestchesscat Aug 18, 2023

#3**+1 **

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.

The3Mathketeers Aug 19, 2023

#3**+1 **

Best Answer

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.

The3Mathketeers Aug 19, 2023