Problem 68 - Algebra 2

Question

Write true or false for each statement. Justify your answer. \(\log _{2} x-4 \log _{2} y=\log _{2} \frac{x}{y^{4}}\)

Step-by-Step Solution

Answer

The statement is True

Step 1: Understanding the logarithmic property

The logarithm property that will be focused on is \(\log_{b}(a*c) = \log_{b}a + \log_{b}c\) if a and c are under multiplication operation within the logarithm and the base is the same (which is 'b' in this case). The reverse is also true that if two logs of the same base are being subtracted, then the values can be divided inside a single log.

Step 2: Applying the property

Applying this property to the given equation \(\log_{2}x - 4\log_{2}y\) can be written as \(\log_{2}x - \log_{2}y^{4}\) as 4 being the multiplier in front of the \(\log_{2}y\) can be taken as the exponent of y based on properties of logarithms.

Step 3: Simplifying the equation

The equation now becomes \(\log_{2}x - \log_{2}y^{4} = \log_{2}\frac{x}{y^{4}}\). Now based on properties of logarithms, the left hand side of the equation simplifies to \(\log_{2}\frac{x}{y^{4}}\) as well.

Step 4: Equating LHS and RHS

Now, both left hand side and right hand side of the equation are equal, which implies that the given statement is true.

Key Concepts

Understanding Logarithms

Logarithms are mathematical tools that help in solving equations involving exponential relationships. They work as the inverse operation of exponentiation. The basic idea is that a logarithm answers the question: "To what power must the base be raised, to produce a given number?" For example, in the equation \(\log_{10}(100) = 2\), it means 10 must be raised to the power of 2 to result in 100.

Logarithms follow several important properties that enable simplification and transformation of expressions:

• Product Rule: \(\log_{b}(MN) = \log_{b}M + \log_{b}N\)
• Quotient Rule: \(\log_{b}(M/N) = \log_{b}M - \log_{b}N\)
• Power Rule: \(\log_{b}(M^n) = n \cdot \log_{b}(M)\)

Understanding and utilizing these rules allows for the conversion of logarithmic expressions into forms that are easier to manipulate and equate. By mastering these principles, you can simplify complex problems into more approachable statements, as shown in the exercise.

Grasping Exponentiation

Exponentiation is the process of raising a number to a power. It is essentially repeated multiplication of a number by itself. The expression \(a^n\) denotes the base 'a' raised to the power 'n'. This process is foundational in numerous mathematical operations and is frequently encountered when working with logarithms.

The rules of exponentiation are tied closely with the properties of logarithms:

• Multiplication: \(a^m \cdot a^n = a^{m+n}\)
• Division: \(a^m / a^n = a^{m-n}\)
• Power of a Power: \((a^m)^n = a^{m \cdot n}\)

Exponentiation and logarithms complement each other. Where exponentiation grows numbers rapidly, logarithms tame these exponential growths into linear scales. This interplay is critical in simplifying equations involving exponential terms, as used in the original exercise for rearranging logarithmic expressions.

Techniques for Simplifying Equations

Simplifying equations involves using algebraic properties to reduce expressions to a more straightforward or more informative form. When dealing with logarithmic expressions, the aim is often to combine or break apart logs using their properties.

In the given problem, simplifying was achieved by using the logarithmic property \(\log_{b}(M/N) = \log_{b}M - \log_{b}N\). Originally, the expression \(\log_{2}x - 4\log_{2}y\) required manipulation. By applying the power law of logarithms, we transformed \(4\log_{2}y\) to \(\log_{2}(y^4)\). This allowed a direct application of the quotient rule to streamline the expression into a simpler form \(\log_{2}\left(\frac{x}{y^4}\right)\).

Utilizing these strategies helps reduce computational complexity, making it easier to verify statements as 'true' or 'false'. Breaking down complex expressions into simpler forms is a valuable technique that aids in problem-solving across various levels of mathematics.