Problem 6 - Algebra and Trigonometry

Question

Evaluate the expression. $$ \log _{12} 9+\log _{12} 16 $$

Step-by-Step Solution

Answer

\( \log_{12} 144 \).

Step 1: Apply the Product Rule of Logarithms

The product rule of logarithms states that \( \log_b (MN) = \log_b M + \log_b N \). Applying this to our expression gives:\[ \log_{12} 9 + \log_{12} 16 = \log_{12} (9 \times 16) \]

Step 2: Identify the operation required

Determine what computation or simplification is needed.

Step 3: Perform the calculation

Execute the computation step by step.

Step 4: Simplify and state the result

Simplify and write the final answer.

Key Concepts

Product Rule of Logarithms

The product rule of logarithms is a powerful tool that simplifies problems involving the addition of logarithms. This rule states that the logarithm of a product is equal to the sum of the logarithms of its factors. In mathematical terms, if you have two numbers, \( M \) and \( N \), the product rule is expressed as \( \log_b (MN) = \log_b M + \log_b N \). This means that multiplying \( M \) and \( N \) together inside the logarithm is equivalent to adding the individual logarithms of \( M \) and \( N \).
For example, in the expression \( \log_{12} 9 + \log_{12} 16 \), by applying the product rule, we can rewrite it as \( \log_{12} (9 \times 16) \). This transformation greatly simplifies the computation, making it easier to evaluate complex logarithmic expressions that would otherwise require several steps.

Logarithmic Expressions

Logarithmic expressions involve operations using logarithms, which are the inverse operations of exponentiation. They appear in various forms depending on what is happening with the numbers inside the logarithm. Logarithms can either be added, subtracted, multiplied, or divided, showing their versatile nature in mathematical expressions.
When you see a logarithmic expression like \( \log_{12} 9 + \log_{12} 16 \), it consists of two separate logarithms being added together. Such an expression suggests a deeper relationship between these two parts, which can often be revealed using rules like the product rule of logarithms. Recognizing these relationships is key to simplifying and solving logarithmic expressions effectively.
Understanding logarithmic expressions is crucial because it helps in breaking down complex problems into simpler parts, making them easier to manage. This approach is commonly used in various fields, including mathematics, science, and engineering, where logs are used regularly to simplify equations and solve real-world problems.

Evaluating Logarithms

Evaluating logarithms means finding the numerical value of a logarithmic expression. It typically involves simplifying the expression as much as possible, using the properties of logarithms, and then calculating the value, often with the help of a calculator when exact numbers are needed.
To evaluate an expression like \( \log_{12} (9 \times 16) \), you would first simplify the expression by calculating \( 9 \times 16 \), which equals 144. Thus, the expression becomes \( \log_{12} 144 \). Now, calculating this logarithm requires looking at how many times you can multiply 12 to get close to 144, which can be further simplified or calculated using logarithm tables or calculators for an exact value.
In practice, evaluating logarithms often combines understanding their properties with calculator use to achieve precise outcomes. Working through these steps not only solidifies comprehension of logarithmic principles but also enhances numerical reasoning skills.