Problem 12 - Precalculus

Question

Expand the given logarithm and simplify. Assume when necessary that all quantities represent positive real numbers. $$ \log _{6}\left(\frac{216}{x^{3} y}\right)^{4} $$

Step-by-Step Solution

Answer

The expanded version is: \( 12 - 12 \cdot \log_{6}(x) - 4 \cdot \log_{6}(y) \).

Step 1: Apply the Power Rule for Logarithms

The power rule for logarithms states that \( \log_b (a^c) = c \cdot \log_b (a) \). Begin by applying this rule to the expression \( \log_{6}\left(\frac{216}{x^{3} y}\right)^{4} \). This expands to: \[ 4 \cdot \log_{6}\left(\frac{216}{x^{3} y}\right). \]

Step 2: Apply the Quotient Rule for Logarithms

The quotient rule for logarithms states that \( \log_b \left(\frac{a}{b}\right) = \log_b(a) - \log_b(b) \). Apply this rule to the expression inside the logarithm: \[ \log_{6}\left(\frac{216}{x^{3} y}\right) = \log_{6}(216) - \log_{6}(x^{3} y). \]

Step 3: Apply the Product Rule for Logarithms

The product rule for logarithms states that \( \log_b(a \cdot b) = \log_b(a) + \log_b(b) \). Apply this rule to \( \log_{6}(x^{3} y) \): \[ \log_{6}(x^{3} y) = \log_{6}(x^{3}) + \log_{6}(y). \]

Step 4: Apply the Power Rule Again

Use the power rule on \( \log_{6}(x^{3}) \): \( \log_{6}(x^{3}) = 3 \cdot \log_{6}(x) \). Substitute back into the expression: \[ \log_{6}(x^{3} y) = 3 \cdot \log_{6}(x) + \log_{6}(y). \]

Step 5: Substitute Back and Simplify the Expression

Substitute \( \log_{6}(x^{3} y) \) back into the equation from Step 2: \[ 4 \cdot (\log_{6}(216) - (3 \cdot \log_{6}(x) + \log_{6}(y))) = 4 \cdot (\log_{6}(216) - 3 \cdot \log_{6}(x) - \log_{6}(y)). \]

Step 6: Distribute the Factor 4

Distribute the factor of 4 across each term in the expression: \[ 4 \cdot \log_{6}(216) - 12 \cdot \log_{6}(x) - 4 \cdot \log_{6}(y). \]

Step 7: Simplify the Logarithmic Expression

Note that \( 216 = 6^3 \). Thus, \( \log_{6}(216) = \log_{6}(6^3) = 3 \cdot \log_{6}(6) = 3 \). Substitute back into the expression: \[ 4 \cdot 3 - 12 \cdot \log_{6}(x) - 4 \cdot \log_{6}(y). \] Hence, simplify to \[ 12 - 12 \cdot \log_{6}(x) - 4 \cdot \log_{6}(y). \]

Key Concepts

Power Rule

The power rule for logarithms is a useful tool when you have an expression where a quantity is raised to an exponent inside the logarithm. The rule states that you can bring down the exponent in front of the logarithm as a multiplier. This is because

• \( \log_b (a^c) = c \cdot \log_b (a) \).

Applying this rule simplifies the expression significantly, as it converts multiplication into a more manageable multiplication of the exponent with the logarithm of the base.
In the given exercise, the expression \( \log_{6}\left(\frac{216}{x^{3} y}\right)^{4} \) utilizes this rule first, allowing us to bring the 4 in front of the logarithm:

• \( 4 \cdot \log_{6}\left(\frac{216}{x^3 y}\right) \)

This transformation makes it easier to further break down the logarithmic expression through other rules.

Quotient Rule

The quotient rule for logarithms allows you to decompose a logarithm involving a fraction into two separate logarithms. This rule states:

• \( \log_b \left(\frac{a}{b}\right) = \log_b(a) - \log_b(b) \).

By applying this rule, the initial logarithmic fraction becomes more straightforward to handle, enabling further simplification.
In our exercise, we apply the quotient rule to \( \log_{6}\left(\frac{216}{x^3 y}\right) \) as follows:

• \( \log_{6}(216) - \log_{6}(x^3 y) \)

Splitting the fraction into separate terms allows us to further break down the second logarithm containing both \(x^3\) and \(y\).
This step-by-step decomposition facilitates the application of other rules like the product rule.

Product Rule

When dealing with logarithms of products, the product rule comes handy by transforming a complex logarithmic expression into a sum of simpler logarithmic terms. The rule is:

• \( \log_b(a \cdot b) = \log_b(a) + \log_b(b) \).

For our exercise, we've previously separated \( \log_{6}(x^3 y) \) using the quotient rule. Now, we apply the product rule to further decompose it:

• \( \log_{6}(x^3 y) = \log_{6}(x^3) + \log_{6}(y) \)

By splitting the terms, you're able to address each part individually, paving the way to apply other rules like the power rule on \( x^3 \).
This transformation is essential for eventually simplifying the entire logarithmic expression.

Simplifying Logarithms

Simplification of logarithmic expressions often involves using a combination of the power, quotient, and product rules repeatedly until no further simplifications are possible.
Our ultimate aim is to express logarithms in their simplest form, making computations less complex.
In the exercise, once all rules have been applied meticulously, the original complex expression \( \log_{6}\left(\frac{216}{x^3 y}\right)^{4} \) is simplified to a much cleaner expression:

• \( 12 - 12 \cdot \log_{6}(x) - 4 \cdot \log_{6}(y) \)

In this case, simplifying involved recognizing that 216 is \(6^3\), allowing substitution back into the expression. It also required understanding how to distribute constants appropriately and making each logarithmic term independently simple. Ultimately, it signifies the power of these logarithm rules in breaking down and simplifying complex expressions effectively.