Question

Which of the following is not equivalent to \(\log _5 \frac{y^4}{3 x}\) ? Justify your answer. (A) \(4 \log _5 y-\log _5 3 x\) (B) \(4 \log _5 y-\log _5 3+\log _5 x\) (C) \(4 \log _5 y-\log _5 3-\log _5 x\) (D) \(\log _5 y^4-\log _5 3-\log _5 x\)

Short answer

The options not equivalent to the original expression \(\log _5 \frac{y^4}{3 x}\) are: (A) \(4 \log _5 y-\log _5 3 x\), (B) \(4 \log _5 y-\log _5 3+\log _5 x\), and (D) \(\log _5 y^4-\log _5 3-\log _5 x\)

Step-by-step solution

Break down the given expression.

Looking at the initial expression: \(\log _5 \frac{y^4}{3 x}\), by using the logarithm properties, it can be rewritten as: \(\log _5 y^4 - \log _5 (3x)\)

Further simplification

The expression: \(\log _5 y^4 - \log _5 (3x)\) can be rewritten again using the logarithm properties: 'logarithm of a product' and 'logarithm of a power'. It simplifies to \( 4 \log _5 y - \log _5 3 - \log _5 x\)

Find the correctly equivalent choice.

After this last simplification, the equivalent options are compared: Option (C) \(4 \log _5 y-\log _5 3-\log _5 x\) is the equivalent.

Identify the non-equivalent options

Comparing the expression with the remaining options, we notice that (A), (B) and (D) are not equivalent to the original expression.

Key concepts

Logarithm Properties

Logarithms are incredibly useful when it comes to simplifying and solving equations. They follow specific properties that allow us to transform expressions in various ways:

• Product Property: \( \log_b (MN) = \log_b M + \log_b N \)
• Quotient Property: \( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \)
• Power Property: \( \log_b (M^p) = p \cdot \log_b M \)

These properties are essential when you want to rewrite logarithmic expressions, like in our exercise, where we simplify \( \log_5 \frac{y^4}{3x} \). They help break down complex expressions into simpler, more manageable parts.

Logarithm of a Product

The logarithm of a product involves breaking down a multiplication within a logarithm into a sum of separate logs. Using the product property \( \log_b (MN) = \log_b M + \log_b N \), we can take a closer look at how this applies to our expressions.

For instance, in the expression \( \log_5 (3x) \), the product property allows us to separate it into:

• \( \log_5 3 + \log_5 x \)

Understanding this property helps in simplifying the original expression by turning the multiplication into addition, making it easier to compare to other logarithmic expressions.

Logarithm of a Power

The power property is another crucial logarithm property that simplifies expressions involving exponents. This property expresses a logarithm of an exponent as a multiple of a logarithm: \( \log_b (M^p) = p \cdot \log_b M \).

In our exercise, when dealing with \( y^4 \), this property helps us transform:

• \( \log_5 y^4 \) into \( 4 \cdot \log_5 y \)

By utilizing this property, we can extract the exponent and multiply it, simplifying the expression and aiding in our further breakdown and comparison to other options given in the problem.

Equivalent Expressions

When solving logarithmic problems, creating equivalent expressions is key. Equivalence means two expressions truly simplify to the same value under all circumstances.

In the exercise, we aimed to find which expressions are equivalent to \( \log_5 \frac{y^4}{3x} \). By applying logarithmic properties, we transformed and compared the provided options:

• Expression (C): \( 4 \log_5 y - \log_5 3 - \log_5 x \) is equivalent.
• Others didn't match the simplified form, as discussed through steps and properties used.

Understanding how the properties work helps you see why certain expressions match or differ, highlighting the value of precise calculations and transformations.