Problem 13 - Elementary and Intermediate Algebra

Question

Match expression with an equivalent expression from the list on the right. \(\log _{3} 10^{11}\) a. \(\frac{\log 11}{\log 3}\) b. \(11 \log _{3} 10\) c. \(\log _{3} 5+\log _{3} 2\) d. \(\log _{3} 10-\log _{3} 11\)

Step-by-Step Solution

Answer

The expression matches option b: \(11 \log _{3} 10\).

Step 1: Understand Properties of Logarithms

One of the properties of logarithms that might be useful for this problem is the power rule given by \( \log_b(x^n) = n \cdot \log_b(x) \). This means you can bring the exponent down in front of the logarithm.

Step 2: Apply the Power Rule

We have the expression \( \log_{3} 10^{11} \). By applying the power rule, this becomes \( 11 \cdot \log_{3} 10 \).

Step 3: Match with Provided Options

From the options given, \( b. \) matches our derived expression. Option \( b. \) is \( 11 \cdot \log_{3} 10 \). The expressions are identical.

Key Concepts

Properties of Logarithms

Logarithms are powerful mathematical tools that help us simplify complex calculations, particularly those involving exponential and multiplication operations. Two key properties that we often use include:

• Product Rule: This states that the logarithm of a product is equal to the sum of the logarithms of each factor. Written mathematically as: \[ \log_b(mx) = \log_b(m) + \log_b(x) \]
• Quotient Rule: This states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator. Mathematically: \[ \log_b\left(\frac{m}{x}\right) = \log_b(m) - \log_b(x) \]

Understanding these properties is vital because they enable us to break down complex expressions into more manageable forms. This simplification is especially useful when calculating logarithms by hand or performing algebraic manipulations. By mastering these properties, we can seamlessly navigate through more challenging logarithmic expressions.

Power Rule in Logarithms

The power rule is one of the most essential properties of logarithms. It helps to simplify expressions where the argument of the logarithm is raised to a power. According to the power rule:\[ \log_b(x^n) = n \cdot \log_b(x) \]This rule is a lifesaver when dealing with logarithms because it allows us to move the exponent in front of the logarithm, converting a power expression into a simple multiplication.
This was the primary approach used in the exercise. By applying the power rule to the expression \( \log_{3} 10^{11} \), it was transformed into \( 11 \cdot \log_{3} 10 \). As shown, what could initially seem daunting simplifies excellently, thanks to the power rule.
Use this rule to make solving logarithmic expressions much more manageable, especially in scenarios involving higher powers or when simplifying equations.

Equivalent Expressions in Logarithms

Often, we encounter multiple expressions that look different but actually represent the same value. Recognizing equivalent expressions is a valuable skill in algebra and logarithms. Equivalent expressions have the same value but may appear different due to applied properties or transformations.
In our exercise, we started with \( \log_{3} 10^{11} \) and converted it using the power rule to \( 11 \cdot \log_{3} 10 \). Both expressions are equal, yet they look distinct initially.
Here are steps to identify equivalent expressions:

• Apply logarithmic properties such as product, quotient, or power rules to simplify or transform the expression.
• Compare transformed expressions to see if they match any given options.
• Remember that equivalent expressions yield the same result if evaluated.

Mastering this helps in solving equations efficiently and enhances your mathematical problem-solving skills.