Problem 53 - Algebra 2

Question

\(\mathrm{ACT} / \mathrm{SAT}\) To what is \(2 \log _{5} 12-\log _{5} 8-\) 2 \(\log _{5} 3\) equal? \(\mathrm{A} \log _{5} 2\) \(\mathrm{B} \log _{5} 3\) \(\mathrm{C} \log _{5} 0.5\) \(\mathrm{D} 1\)

Step-by-Step Solution

Answer

The expression simplifies to \( \log_{5} 2 \), which is option A.

Step 1: Apply Logarithm Power Rule

Begin by applying the power rule of logarithms, which states that \( a \cdot \log_b c = \log_b (c^a) \). For the term \(2 \, \log_{5} 12\), rewrite it as \(\log_{5} (12^2)\). Similarly, for the term \(2 \, \log_{5} 3\), rewrite it as \(\log_{5} (3^2)\). This gives us: \[ \log_{5} (12^2) - \log_{5} 8 - \log_{5} (3^2) \] which simplifies to: \[ \log_{5} 144 - \log_{5} 8 - \log_{5} 9 \]

Step 2: Use the Quotient Rule

The quotient rule of logarithms states that \( \log_b a - \log_b c = \log_b \left( \frac{a}{c} \right) \). Apply this to \( \log_{5} 144 - \log_{5} 8 \):\[ \log_{5} \left( \frac{144}{8} \right) \]This simplifies to:\[ \log_{5} 18 \] Now apply quotient rule again with \( \log_{5} 18 - \log_{5} 9 \):\[ \log_{5} \left( \frac{18}{9} \right) \]This simplifies to:\[ \log_{5} 2 \]

Step 3: Compare with Answer Choices

The simplified expression is \( \log_{5} 2 \), which matches with option A in the answer choices provided.

Key Concepts

Logarithm Power Rule

Understanding how to manipulate logarithms is crucial in algebra, especially with the power rule of logarithms. This rule allows you to bring constants in front of a logarithm inside, transforming the expression. More specifically, the power rule states that if you have a term like \( a \cdot \log_b c \), you can rewrite it as \( \log_b (c^a) \). This is incredibly useful for simplifying expressions in equations.
For instance, in the problem given, you have \(2 \log_{5} 12\). By applying the power rule:

• Use the constant 2 as an exponent of 12 inside the logarithm.
• The expression converts to \(\log_{5} (12^2)\).

This transformation helps simplify complex expressions, making it easier to work with logarithms in algebraic contexts.

Quotient Rule of Logarithms

In algebra, the quotient rule of logarithms is another powerful tool for simplifying equations. It states that the difference between two logarithms of the same base can be expressed as the logarithm of their quotient. Mathematically, \( \log_b a - \log_b c = \log_b \left( \frac{a}{c} \right) \).
In the given exercise, after using the power rule, the expression needed more simplification:

• First, you subtract \( \log_{5} 8 \) from \( \log_{5} 144 \), resulting in \( \log_{5} \left( \frac{144}{8} \right) = \log_{5} 18 \).
• Next, you subtract \( \log_{5} 9 \), leading to \( \log_{5} \left( \frac{18}{9} \right) = \log_{5} 2 \).

Using the quotient rule effectively reduces the complexity of the expressions, making the problem-solving process smoother and more intuitive.

Algebraic Expressions

Algebraic expressions form the building blocks of equations in mathematics. They involve numbers, variables, and arithmetic operations. In our context, logarithms are a part of these expressions, acting as unique functions that simplify multiplicative processes.
When dealing with logarithms in algebraic expressions, understanding their properties is crucial. The transformation of terms using the power rule and quotient rule refines expressions, leading to solutions. This manipulation of expressions is a fundamental aspect of algebra, enabling easier solution derivation.
These expressions transform based on provided rules, achieving a simplified form that mirrors initial configurations like in the given problem, which ultimately simplifies to \( \log_{5} 2 \). Keeping a firm grasp on managing algebraic expressions helps in achieving a deeper understanding and in solving equations efficiently.