Problem 19 - College Algebra

Question

For the following exercises, use the properties of logarithms to expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs. $$ \log \left(x^{2} y^{3} \sqrt[3]{x^{2} y^{5}}\right) $$

Step-by-Step Solution

Answer

\( \frac{8}{3}\log(x) + \frac{14}{3}\log(y) \)

Step 1: Apply the Property of Products in Logs

The logarithm of a product can be expressed as the sum of the logarithms. Apply this property to the expression: \( \log(x^2 y^3 \sqrt[3]{x^2 y^5}) = \log(x^2) + \log(y^3) + \log(\sqrt[3]{x^2 y^5}) \).

Step 2: Simplify the Power of Each Variable

Use the power rule of logarithms, which states \( \log(a^b) = b \cdot \log a \), to simplify each term: \( \log(x^2) = 2\log(x) \) and \( \log(y^3) = 3\log(y) \).

Step 3: Simplify the Radical Term Using Logs

The expression inside the cube root can also be expanded. Replace the cube root in the logarithm: \( \log(\sqrt[3]{x^2 y^5}) = \log((x^2 y^5)^{1/3}) \). This can be further simplified to: \( \frac{1}{3} \cdot \log(x^2 y^5) \).

Step 4: Apply Product Rule to the Cube Root Term

Now apply the logarithm product rule to \( \log(x^2 y^5) \): \( \log(x^2 y^5) = \log(x^2) + \log(y^5) \), so \( \frac{1}{3}(2\log(x) + 5\log(y)) = \frac{2}{3}\log(x) + \frac{5}{3}\log(y) \).

Step 5: Combine All Terms

Combine all the expanded terms from previous steps: \( 2\log(x) + 3\log(y) + \frac{2}{3}\log(x) + \frac{5}{3}\log(y) \).

Step 6: Simplify the Logarithmic Expression

Combine like terms: For \( \log(x) \) terms: \( 2\log(x) + \frac{2}{3}\log(x) = \frac{6}{3}\log(x) + \frac{2}{3}\log(x) = \frac{8}{3}\log(x) \). For \( \log(y) \) terms: \( 3\log(y) + \frac{5}{3}\log(y) = \frac{9}{3}\log(y) + \frac{5}{3}\log(y) = \frac{14}{3}\log(y) \).

Step 7: Conclusion: Final Expanded Logarithmic Expression

The expanded form of the given logarithmic expression is \( \frac{8}{3}\log(x) + \frac{14}{3}\log(y) \).

Key Concepts

Properties of Logarithms

Logarithms have several important properties that simplify complex expressions. These properties make dealing with logarithmic equations more manageable, especially when expanding or condensing expressions. Here are the key properties:

• **Product Property:** The logarithm of a product is the sum of the logarithms of the factors. Formally, \(\log(a \cdot b) = \log(a) + \log(b)\).
• **Quotient Property:** The logarithm of a quotient is the difference between the logarithms of the numerator and the denominator: \(\log\left(\frac{a}{b}\right) = \log(a) - \log(b)\).
• **Power Property:** This property states that the logarithm of a number raised to an exponent is the exponent times the logarithm of the number: \(\log(a^b) = b \cdot \log(a)\).

These properties are fundamental to understanding how to manipulate logarithmic expressions. They allow us to break down complicated expressions into simpler, more manageable components. For example, in the given exercise, we initially applied the product property to separate each component of the product before further simplifying with other properties.

Power Rule of Logarithms

The power rule of logarithms is a powerful tool that simplifies expressions where logarithms are involved with exponents. This rule is articulated as \(\log(a^b) = b \cdot \log(a)\).
This means whenever you encounter a logarithm with an exponent, you can "bring the exponent out front" as a multiplier of the logarithm. This property helps simplify complex expressions by turning products inside the logarithm into sums outside it.
For instance, in our exercise, we have terms like \(x^2\) and \(y^3\). By applying the power rule, these turn into \(2 \cdot \log(x)\) and \(3 \cdot \log(y)\), respectively. This simplification allows further manipulation and combination of terms.
Whenever you're faced with logarithms involving exponents, remember the power rule—it makes life much easier by reducing the complexity of the expression.

Product Rule of Logarithms

The product rule is essential for splitting up logarithmic expressions of products into individual terms. According to this property, the logarithm of a product equals the sum of the logarithms of its factors: \(\log(a \cdot b) = \log(a) + \log(b)\).
Using this rule allows us to take a single logarithm of a product and separate it into more straightforward components. This is especially useful in cases where each part of the product has further manipulations available, like applying the power rule.
In this exercise, we initially have the expression \(\log(x^2 y^3 \sqrt[3]{x^2 y^5})\). By the product rule, we break it into \(\log(x^2) + \log(y^3) + \log(\sqrt[3]{x^2 y^5})\). Each term is then easier to handle separately, notably employing the power rule where applicable.
By comprehending the product rule, students can manage complex products in logarithmic forms by isolating factors, simplifying each, and eventually recombining them for complete expansion.