Problem 516 - Precalculus

Question

Rewrite \(-\log _{y}\left(\frac{1}{12}\right)\) as a single logarithm.

Step-by-Step Solution

Answer

\(\log_{y}(12)\)

Step 1: Apply Logarithm Property

Apply the property that states \(-\log_b(x) = \log_b(x^{-1})\). This means we can rewrite the negative sign in front of the log in terms of an exponent: \(-\log_{y}\left(\frac{1}{12}\right) = \log_{y}\left(\left(\frac{1}{12}\right)^{-1}\right)\).

Step 2: Simplify the Expression

Rewrite \(\left(\frac{1}{12}\right)^{-1}\) using the property of exponents that states \(\left(\frac{a}{b}\right)^{-1} = \frac{b}{a}\). Here, this becomes \(\frac{12}{1}\), or simply 12. Thus, the expression becomes:\(\log_{y}(12)\).

Key Concepts

The Property of Logarithms

To understand the process of rewriting logarithmic expressions, we need to start with some basic properties of logarithms. Specifically, there's an important property which helps manage negative signs. This property says that for any logarithm, a negative sign in front of it can be represented as an inverse power. In technical terms:

• If you have \(-\log_b(x)\), this can be rewritten as \(\log_b(x^{-1})\).

This means you can change a negative logarithm into one involving an inverse exponent.
In our exercise, we dealt directly with this property by transforming \(-\log_{y}(\frac{1}{12})\) into \(\log_{y}(\frac{1}{12})^{-1}\). This is a fundamental step to understand since it sets the stage for making expressions more manageable. Remember, when you see a negative logarithm, think about flipping the expression into an inverse.

Unpacking Negative Exponents

Negative exponents can be puzzling at first, but they follow straightforward rules. The basic idea is that a negative exponent indicates that a number should be taken as a reciprocal. Here's how you can break it down:

• When you have \(x^{-1}\), it is the same as saying \(\frac{1}{x}\).
• For \(\left(\frac{a}{b}\right)^{-1}\), it becomes \(\frac{b}{a}\).

This means that applying a negative exponent to a fraction like \(\left(\frac{1}{12}\right)^{-1}\) results in flipping the fraction to \(\frac{12}{1} = 12\).
In our exercise, recognizing that the fraction's reciprocal can be simplified in this way helps transition from negative exponents, transforming the expression into something easier to handle. Steps like this streamline the process, allowing you to swiftly move from complex expressions to their simplified forms.

Steps to Simplify Expressions

Simplifying expressions makes them easier to interpret and work with. Sometimes expressions can seem complex, but breaking them down simplifies the task. This involves:

• Looking for opportunities to apply known mathematical properties.
• Removing unnecessary complications.

In this context, simplifying meant applying a property to clear the way for an easier equivalent: \(\left(\frac{1}{12}\right)^{-1} = 12\).
So, our original expression \(-\log_{y}\left(\frac{1}{12}\right)\) was simplified ultimately to \(\log_{y}(12)\). Each step eradicated complexity, leaving us with a cleaner, more understandable expression. Simplification isn't only about shorter expressions; it's about transforming them to reveal clearer, more accessible meanings.