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Given log53 = 0.6826 and log58=1.2920, evaluate the expressions.

(a) log52

(b) log5(75/8)

Aug 18, 2023

#3
+177
+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.

Aug 19, 2023

#3
+177
+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