Problem 22 - College Algebra

Question

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{b}\left(x y^{3}\right) $$

Step-by-Step Solution

Answer

The logarithmic expression \(\log _{b}\left(x y^{3}\right)\) expands to \( \log_b(x) + 3\log_b(y)\)

Step 1: Apply Logarithmic Multiplication Property

Observe that the argument of the logarithm, \(xy^{3}\), can be expressed as a product. The multiplication property of logarithms states that the logarithm of a product is the sum of the logarithms, so express \( \log _{b}\left(x y^{3}\right)\) as \( \log_b(x) + \log_b(y^{3})\).

Step 2: Apply Logarithmic Power Rule

Notice that the second term in the sum, \(\log_b(y^{3})\), involves an exponent inside the argument of the log. The rule for this situation states that \(\log_bM^n = n \cdot \log_bM\). Rewrite this term as \(3 \cdot \log_b(y)\) so the expanded expression becomes \( \log_b(x) + 3\log_b(y)\) .

Key Concepts

Logarithmic Multiplication Property

The logarithmic multiplication property is a vital tool for expanding logarithmic expressions. It allows us to transform the logarithm of a product into a sum of logarithms. This aids in simplifying and understanding complex logarithmic expressions. For example, if you have an expression like \(\log_b(xy)\), you're looking at the logarithm of the product \(xy\). According to this property, you can rewrite it as \(\log_b(x) + \log_b(y)\).

Why is this useful? Because the logarithm of a product means dealing with both \(x\) and \(y\) together, which can sometimes be more complex than handling them separately. Converting the product into a sum makes it easier to manipulate and evaluate each part individually.

• Helps break down complex expressions
• Simplifies calculation
• Essential for further expansion

Applying this property is like opening a locked box and inspecting each item separately, making the problem much easier to handle.

Logarithmic Power Rule

The logarithmic power rule is another important technique when dealing with logarithmic expressions. This rule helps to simplify expressions where logarithms contain powers. Suppose you have an expression such as \(\log_b(y^3)\).

Utilizing the logarithmic power rule, which states \(\log_b(M^n) = n \cdot \log_b(M)\), you can rewrite \(\log_b(y^3)\) as \(3 \cdot \log_b(y)\).

• Turns powers inside logs into multipliers
• Simplifies the complexity of expressions
• Makes further expansion easier to manage

By moving the power 3 to the front, you end up with a simpler linear expression. This transformation is crucial, especially in calculations where you need to solve equations or further expand the logarithmic terms. Handling the power this way prepares components like \(\log_b(y)\) to be manipulated independently and more straightforwardly in subsequent steps.

Expanding Logarithmic Expressions

Expanding logarithmic expressions is a process of applying known logarithmic properties, like multiplication and power rules, to rewrite expressions in simpler or more usable forms. This is particularly useful in simplifying complex logarithmic equations or in solving problems where evaluating the expression directly might be challenging.

For the expression \(\log_b(xy^3)\), by first applying the multiplication property, it becomes \(\log_b(x) + \log_b(y^3)\). Next, using the power rule on \(\log_b(y^3)\), it simplifies to \(\log_b(x) + 3 \cdot \log_b(y)\). This final form is much simpler to analyze and use.

• Transforms expressions to simpler forms
• Aids in problem-solving and equation solving
• Helps in both manual and theoretical calculations

Expanding is like taking a complex piece of text and translating it into clear, plain language. Each part becomes easier to understand, evaluate, and if necessary, to further manipulate depending on the problem at hand. This strategy is widely applicable in both academic contexts and practical applications.