Use the change-of-base theorem to find an approximation to four decimal places for each logarithm. See Example 8. . log_1/2 3

Verified step by step guidance

1. Identify the change-of-base formula: \( \log_b a = \frac{\log_c a}{\log_c b} \), where \( c \) is a new base, typically 10 or \( e \).
2. Apply the change-of-base formula to \( \log_{1/2} 3 \): \( \log_{1/2} 3 = \frac{\log_{10} 3}{\log_{10} (1/2)} \).
3. Use a calculator to find \( \log_{10} 3 \) and \( \log_{10} (1/2) \).
4. Divide the result of \( \log_{10} 3 \) by the result of \( \log_{10} (1/2) \).
5. Round the final result to four decimal places.

Key Concepts

Change of Base Theorem

The Change of Base Theorem allows us to compute logarithms with bases other than 10 or e by converting them into a more manageable form. Specifically, it states that for any positive numbers a, b, and c (where a ≠ 1 and b ≠ 1), the logarithm can be expressed as log_b(a) = log_c(a) / log_c(b). This theorem is particularly useful when using calculators that only support certain bases.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and are defined as log_b(a) = c if and only if b^c = a. They are essential in solving equations involving exponentials and are widely used in various fields, including science and engineering. Understanding the properties of logarithms, such as the product, quotient, and power rules, is crucial for manipulating and simplifying logarithmic expressions.

Approximation and Decimal Places

When calculating logarithms, especially with the change of base theorem, approximations are often necessary. This involves rounding the result to a specified number of decimal places, which is important for precision in applications. In this context, finding an approximation to four decimal places means ensuring that the result is accurate to the fourth digit after the decimal point, which can affect the outcome in practical scenarios.