Use the change-of-base theorem to find an approximation to four decimal places for each logarithm. See Example 8. log_√13 12

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. In this problem, \( b = \sqrt{13} \) and \( a = 12 \).
3. Choose a common base for the logarithms, such as base 10 (common logarithm) or base \( e \) (natural logarithm).
4. Apply the change-of-base formula: \( \log_{\sqrt{13}} 12 = \frac{\log_{10} 12}{\log_{10} \sqrt{13}} \) or \( \log_{\sqrt{13}} 12 = \frac{\ln 12}{\ln \sqrt{13}} \).
5. Use a calculator to find \( \log_{10} 12 \) and \( \log_{10} \sqrt{13} \) or \( \ln 12 \) and \( \ln \sqrt{13} \), then divide the results to get the approximation.

Key Concepts

Change of Base Theorem

The Change of Base Theorem allows us to convert logarithms from one base to another. It states that for any positive numbers a, b, and c (where a and b are not equal to 1), log_b(c) can be expressed as log_a(c) / log_a(b). This theorem is particularly useful when calculating logarithms with bases that are not easily computable using standard calculators.

Logarithm Properties

Logarithms have several key properties that simplify calculations. For instance, the product property states that log_b(mn) = log_b(m) + log_b(n), while the quotient property states that log_b(m/n) = log_b(m) - log_b(n). Understanding these properties is essential for manipulating logarithmic expressions and solving logarithmic equations effectively.

Approximation Techniques

When calculating logarithms, especially with non-integer bases or arguments, approximation techniques may be necessary. This often involves using a calculator to find logarithmic values to a specified number of decimal places. In this context, approximating log_√13(12) requires applying the Change of Base Theorem and then using a calculator to achieve the desired precision.