Problem 47 - Algebra A Combined Function

Question

Write each expression as a sum or difference of logarithms. Assume that variables represent positive numbers. $$ \log _{6} x^{4} y^{5} $$

Step-by-Step Solution

Answer

\(4 \log_6 x + 5 \log_6 y\)

Step 1: Apply the Power Rule of Logarithms

The power rule states that \( \log_b (m^n) = n \cdot \log_b (m) \). Apply this rule separately for the terms \( x^4 \) and \( y^5 \) inside the logarithm:\(\log_6 (x^4 y^5) = \log_6 (x^4) + \log_6 (y^5)\)Then apply the power rule:\(= 4\log_6 (x) + 5\log_6 (y)\)

Step 2: Express Each Term as a Sum of Logarithms

From Step 1, you already have the expression split into two parts:\(4\log_6 (x) + 5\log_6 (y)\)No further simplification is needed, as this expression is already a sum of logarithms.

Key Concepts

Power Rule of Logarithms

The power rule of logarithms is a handy tool when dealing with expressions where a number, variable, or term is raised to a power inside the logarithm. When you have an expression like \( \log_b (m^n) \), the power rule allows you to simplify it by bringing the exponent \( n \) to the front as a multiplicative factor: \[ \log_b (m^n) = n \cdot \log_b (m) \] This means that instead of managing a compound term within the logarithm, you simplify it into a product that is easier to work with. For instance, if you have \( \log_6 (x^4) \), applying the power rule you get:

• \(4 \cdot \log_6 (x)\)

This principle applies not only to numeric terms but also to algebraic variables, making complex logarithmic expressions more manageable and clearer.

Logarithm Properties

Understanding the fundamental properties of logarithms can make manipulating expressions much more straightforward. There are several core properties that can be applied when simplifying or expanding logarithmic expressions.

• Product Property: \( \log_b (mn) = \log_b (m) + \log_b (n) \) --> If you have a product inside a logarithm, it can be split into a sum of two separate logarithms.
• Quotient Property: \( \log_b \left(\frac{m}{n}\right) = \log_b (m) - \log_b (n) \) --> A division inside a logarithm becomes a subtraction.
• Power Property: Discussed earlier, \( \log_b (m^n) = n \cdot \log_b (m) \).

These properties are derived from the inverse relationship between logarithms and exponentials, providing a foundational framework for converting between multiplication/division and addition/subtraction in logarithmic forms. By mastering these, you can break down complex expressions step-by-step.

Sum and Difference of Logarithms

In many mathematical problems, such as the original exercise, you may need to express logarithmic terms as a sum or difference of logarithms. This technique is crucial when you want to solve equations or simplify expressions further. The key is to recognize when to employ the product and quotient properties:

• With sums, apply the product property to condense or expand: \( \log_b (mn) \rightarrow \log_b (m) + \log_b (n) \).
• With differences, use the quotient property: \( \log_b \left(\frac{m}{n}\right) \rightarrow \log_b (m) - \log_b (n) \).

These transformations preserve the equality of the expression while altering its appearance to be more convenient for further computation or interpretation. For example, in the logarithm \( \log_6 (x^4 y^5) \), we first handle the power rules to break it down into \(4\log_6 (x) + 5\log_6 (y)\), effectively expressing it as a sum that might be more practical for additional steps or theoretical explanations.