Use the change-of-base theorem to find an approximation to four decimal places for each logarithm. See Example 8. log_√19 5

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 scientific calculator or logarithm tables to find values to a specified number of decimal places. In this context, approximating log_√19(5) requires careful application of the Change of Base Theorem and possibly numerical methods to achieve the desired precision.