Use a calculator to find an approximation to four decimal places for each logarithm. log_2/3 5/8

Verified step by step guidance

1. Recognize that the logarithm given is \( \log_{\frac{2}{3}} \frac{5}{8} \), which means the logarithm of \( \frac{5}{8} \) with base \( \frac{2}{3} \).
2. Recall the change of base formula for logarithms: \( \log_a b = \frac{\log_c b}{\log_c a} \), where \( c \) is any positive number (commonly 10 or \( e \) for calculator use).
3. Apply the change of base formula using base 10 (common logarithm): \[ \log_{\frac{2}{3}} \frac{5}{8} = \frac{\log_{10} \left( \frac{5}{8} \right)}{\log_{10} \left( \frac{2}{3} \right)} \].
4. Use a calculator to find the values of \( \log_{10} \left( \frac{5}{8} \right) \) and \( \log_{10} \left( \frac{2}{3} \right) \) separately, making sure to keep enough decimal places for accuracy.
5. Divide the two logarithm values obtained in the previous step and round the result to four decimal places to get the final approximation.

Key Concepts

Logarithms and Their Bases

A logarithm answers the question: to what power must the base be raised to produce a given number? In this problem, the base is a fraction (2/3), and the argument is another fraction (5/8). Understanding how logarithms work with fractional bases and arguments is essential for solving the problem.

Change of Base Formula

The change of base formula allows you to compute logarithms with any base using a calculator that typically only has log base 10 or natural log (ln). It states that log_b(a) = log(a) / log(b), where log can be any logarithm base available on the calculator. This formula is crucial for approximating log base 2/3 of 5/8.

Rounding and Decimal Approximation

After calculating the logarithm value, it is important to round the result to the specified number of decimal places, here four. Proper rounding ensures the answer is both accurate and presented in a standard format, which is important for clarity and correctness in mathematical communication.