Question

Match the expression with the logarithm that has the same value. Justify your answer. A. \(\log _3 64\) B. \(\log _3 3\) C. \(\log _3 12\) D. \(\log _3 36\) $$6 \log _3 2$$

Short answer

The logarithmic expression \(6 \log _3 2\) matches choice A (\(\log _3 64\)).

Step-by-step solution

Apply logarithmic rules

The power rule states that you can bring powers in front of the log. Therefore, \(6 \log _3 2\) simplifies to \(\log _3 2^6\).

Simplify the expression

Calculate \(2^6\) to get 64. As a result, \(6 \log _3 2\) simplifies to \(\log _3 64\).

Match the expression

Compare this simplified expression to the available choices. \(\log _3 64\) matches choice A. Hence, \(\log _3 64 = 6 \log _3 2\).

Key concepts

Logarithm Properties

Logarithms have unique characteristics that make them invaluable in mathematics, especially when dealing with exponential relationships. Understanding logarithmic properties helps to simplify complex expressions and solve equations that would otherwise be difficult to manage. The two fundamental properties that provide the backbone for other properties are:

• The Product Rule: The logarithm of a product is the sum of the logarithms of the factors. In mathematical terms, this is expressed as \( \log_b (MN) = \log_b M + \log_b N \).
• The Quotient Rule: The logarithm of a quotient is the difference of the logarithms of the numerator and the denominator, represented as \( \log_b (\frac{M}{N}) = \log_b M - \log_b N \).

These properties are the reason why logarithms can transform multiplication and division into addition and subtraction, greatly simplifying the process of dealing with exponents.

Power Rule for Logarithms

When it comes to the power rule for logarithms, it's a straightforward yet powerful property that relates exponentiation to multiplication. It states that the logarithm of a number raised to an exponent is the exponent times the logarithm of the number. Mathematically, this is written as \( \log_b (M^p) = p \cdot \log_b M \).

In simpler terms, if you have a logarithm of an exponentiated number, you can

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions involves using the properties of logarithms to make them easier to understand or solve. In the given exercise, the simplification process shows the power rule in action. Here's a general guide to simplify such expressions:

1. Identify and apply the appropriate logarithmic rule, like the power rule, product rule, or quotient rule, depending on the expression.
2. Execute any arithmetic needed, such as calculating the value of an exponent or consolidating log terms.
3. Match the simplified expression to other logarithmic forms to find equivalencies or solve equations.

By following these steps, expressions that may seem complex at first can be transformed into a more manageable form. It's crucial to be fluent in these techniques to handle more challenging logarithmic problems and to deepen the understanding of how logarithms operate within the broader context of algebra and calculus.