Use the change-of-base theorem to find an approximation to four decimal places for each logarithm. See Example 8. log_2 5

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 log_b(a) can be expressed as log_k(a) / log_k(b) for any positive k. This is particularly useful when using calculators that typically only compute common (base 10) or natural (base e) logarithms.

Logarithm Basics

A logarithm is the inverse operation to exponentiation, answering the question: to what exponent must a base be raised to produce a given number? For example, log_b(a) = c means that b^c = a. Understanding the properties of logarithms, such as the product, quotient, and power rules, is essential for manipulating and solving logarithmic expressions.

Approximation Techniques

When calculating logarithms, especially with the Change of Base Theorem, approximation techniques may be necessary to achieve a desired level of precision. This often involves using a calculator to find the values of the logarithms in the numerator and denominator, and then performing the division to obtain the final result. Rounding to four decimal places ensures that the answer is both precise and practical for applications.