Problem 69 - Intermediate Algebra

Question

For Problems \(69-80\), express each of the following as the sum or difference of simpler logarithmic quantities. Assume all variables represent positive real numbers. For example, $$ \begin{aligned} \log _{b} \frac{x^{3}}{y^{2}} &=\log _{b} x^{3}-\log _{b} y^{2} \\ &=3 \log _{b} x-2 \log _{b} y \end{aligned} $$ $$ \log _{b} x y z $$

Step-by-Step Solution

Answer

\(\log_b (x y z) = \log_b x + \log_b y + \log_b z.\)

Step 1: Identify the Logarithm Rule

The given problem is to express \( \log_b (x y z) \) as a sum of simpler logarithmic terms. We will use the property of logarithms that states \( \log_b (MN) = \log_b M + \log_b N \). This property helps in separating the logarithm of a product into the sum of logarithms.

Step 2: Apply the Product Rule

Apply the logarithm property \( \log_b (MN) = \log_b M + \log_b N \) to break down \( \log_b (x y z) \). This gives us: \[ \log_b (x y z) = \log_b x + \log_b y + \log_b z \].

Step 3: Simplify the Expression

Since there are no more complex products to decompose using the product rule, the expression \( \log_b x + \log_b y + \log_b z \) is already in its simplest form.

Key Concepts

Logarithm Rules

Logarithm rules are fundamental properties that allow us to manipulate and simplify logarithmic expressions. These rules are powerful tools in mathematics, providing ways to transform complicated log expressions into simpler forms. The primary rules include:

• Product Rule: \(\log_b (MN) = \log_b M + \log_b N\)
• Quotient Rule: \(\log_b \frac{M}{N} = \log_b M - \log_b N\)
• Power Rule: \(\log_b M^n = n \log_b M\)

These rules allow us to decompose logarithmic expressions involving products, divisions, or powers into sums, differences, or scaled terms of logarithms. For example, if given a complex product inside a logarithm, the product rule enables us to express it as the sum of individual logarithms. This concept not only aids in solving mathematical problems but also provides a clearer understanding of logarithmic relationships.

Product Rule

The product rule of logarithms is one of the key tools used to simplify expressions involving the logarithm of a product. According to this rule, the logarithm of a product is equal to the sum of the logarithms of its factors.To elaborate, if you have an expression like \(\log_b (xy)\), the product rule lets you transform it into \(\log_b x + \log_b y\).
Let's see how it applies:

• Given \(\log_b (xyz)\), using the product rule, it becomes \(\log_b x + \log_b y + \log_b z\).
• This transformation is very useful when dealing with products inside logarithms, as it allows you to work with simpler, individual logarithmic expressions.

By focusing on the components of the product separately, this rule makes complex expressions easier to handle and solve.

Simplifying Logarithms

Simplifying logarithms involves using the rules of logarithms to break down or aggregate terms into their simplest form. This process is essential for solving logarithmic equations and for simplifying more complex algebraic expressions that involve logarithms.In the given example, simplifying started by applying logarithm rules:

• Identify complex terms, such as products, that can be rewritten using logarithm properties.
• Apply the suitable logarithm rule, like the product rule, to decompose the expression.

For instance, an expression like \(\log_b (xyz)\) might look simple at first, but it consists of three components: \(x, y, \) and \(z\). By using the product rule, you can break it down to \(\log_b x + \log_b y + \log_b z\).
This step-by-step approach not only makes an expression simpler but also makes it easier to compute or further manipulate. Simplifying logarithms is an important skill, especially in calculus and algebra, as it often aids in integral and derivative computations involving logarithmic functions.