91Ó°ÊÓ

Logarithmic expressions are a way to represent the power to which a number, known as the base, must be raised to obtain another number. For example, the expression \(\log_b(x)\) answers the question: 'What power do we need to raise b to get x?'. In our exercise, we deal with the natural logarithm, denoted as \(\ln\), with the base \(e\) (Euler's number, approximately 2.71828).

Working with logarithmic expressions often requires expanding or condensing them, which in turn involves a solid understanding of logarithmic properties such as the product rule, quotient rule, and power rule. Expanding a logarithm makes complex calculations easier and allows us to solve for variables that may initially be locked within the log.

The quotient rule of logarithms is a transformative property that simplifies the division of two numbers within a logarithm into the subtraction of two logarithms. Generally, for any positive numbers \(a\) and \(b\) and base \(c\), it is defined as \(\log_c(\frac{a}{b}) = \log_c(a) - \log_c(b)\). In our exercise, this property is used to separate the expression involving division (\(\ln(\frac{x y}{z})\)) into the subtraction of two logs:

• First, you express \(\ln(\frac{x y}{z})\) as \(\ln(x y) - \ln(z)\).
• By splitting the logarithm of a quotient into the difference of two logs, you're setting the stage for further expansion using the product rule.

Like the quotient rule, the product rule of logarithms is a powerful tool for manipulating expressions. For any positive numbers \(a\) and \(b\) and base \(c\), the rule states that \(\log_c(ab) = \log_c(a) + \log_c(b)\).

After applying the quotient rule, the exercise leads us to \(\ln(x) + \ln(y) - \ln(z)\), where the expression \(\ln(x y)\) has been divided into \(\ln(x) + \ln(y)\) using the product rule. This step simplifies the logarithm of a product into a sum of individual logs and is particularly useful when further calculations or differentiations are required.

Example Use in Simplifying Expressions

Considering a product inside a natural logarithm like \(\ln(x y)\), applying the product rule makes it \(\ln(x) + \ln(y)\), breaking it down into simpler parts that can subsequently be worked with individually or even evaluated if \(x\) and \(y\) values are known.