Problem 49 - Essential Calculus Early Transcendentals

Question

Use the properties of logarithms to expand the quantity. $$ \ln \frac{x^{2}}{y^{3} z^{4}} $$

Step-by-Step Solution

Answer

\(2 \ln x - 3 \ln y - 4 \ln z\)

Step 1: Apply the Quotient Rule

The first step in expanding a logarithmic expression is to use the quotient rule, which states that the logarithm of a quotient is the difference of the logarithms of the numerator and the denominator. For our expression, this step results in:\[ \ln \left( \frac{x^2}{y^3 z^4} \right) = \ln (x^2) - \ln (y^3 z^4) \]

Step 2: Apply the Product Rule

Next, we further expand the denominator using the product rule for logarithms. The rule states that the logarithm of a product is the sum of the logarithms of the factors. Applying this rule, we expand \( \ln (y^3 z^4) \):\[ \ln (y^3 z^4) = \ln (y^3) + \ln (z^4) \]

Step 3: Substitute the Expanded Denominator

Substitute the expanded form of the denominator from Step 2 back into the expression we obtained in Step 1:\[ \ln (x^2) - (\ln (y^3) + \ln (z^4)) \]

Step 4: Apply the Power Rule

Finally, use the power rule of logarithms, which states that \( \ln (a^b) = b \ln a \), to expand all terms fully:\[ \ln (x^2) = 2 \ln x, \quad \ln (y^3) = 3 \ln y, \quad \ln (z^4) = 4 \ln z \]

Step 5: Combine Everything

Substitute these expanded terms back into the expression:\[ 2 \ln x - (3 \ln y + 4 \ln z) \] Simplify the expression inside the parentheses:\[ 2 \ln x - 3 \ln y - 4 \ln z \]

Key Concepts

Quotient Rule

The quotient rule for logarithms is an important property that simplifies expressions involving division. According to this rule, when you have a logarithm of a quotient, you can break it into the difference between two separate logarithms. This can be particularly useful when simplifying or expanding logarithmic expressions. For example, let’s think about the expression \( \ln \left( \frac{a}{b} \right) \). The quotient rule tells us:

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

In other words, we subtract the logarithm of the denominator from the logarithm of the numerator. Remember, this rule is applicable for any positive numbers \( a \) and \( b \). In our original problem, applying the quotient rule was the first step:

• The logarithm of \( \frac{x^2}{y^3 z^4} \) becomes \( \ln(x^2) - \ln(y^3 z^4) \)

By breaking it down this way, it sets the stage for using other logarithmic rules to further simplify or expand the expression.

Product Rule

The product rule for logarithms states that the logarithm of a product equals the sum of the logarithms of the factors. This property is useful for expanding expressions involving multiplication. It allows you to see each component separately within a logarithm. Consider an expression like \( \ln(ab) \). According to the product rule:

• \( \ln(ab) = \ln a + \ln b \)

This suggests that when you encounter multiplication within a logarithm, you can handle each factor independently. In our exercise, we applied the product rule to the term \( \ln(y^3 z^4) \):

• \( \ln(y^3 z^4) \) expands to \( \ln(y^3) + \ln(z^4) \)

This deconstruction makes it easier to apply other rules, like the power rule, to further simplify the expression. It's important to ensure that each fator inside the logarithm remains positive for the rule to hold.

Power Rule

The power rule for logarithms allows you to simplify expressions where a logarithm has an exponent inside of it. It states that the logarithm of a number raised to a power is the exponent times the logarithm of the number. For instance, if you have \( \ln(a^b) \), the power rule provides:

• \( \ln(a^b) = b \ln a \)

This property is handy when dealing with exponential logarithmic terms because it brings the exponent down to the front, making the expression easier to compute and understand. In the given exercise, we applied the power rule to fully expand the logarithmic terms:

• \( \ln(x^2) = 2 \ln x \)
• \( \ln(y^3) = 3 \ln y \)
• \( \ln(z^4) = 4 \ln z \)

This transformation is crucial in simplifying the original expression. It reduces the complexity by linearizing the exponential components inside each logarithm. By understanding and applying the power rule, students can confidently expand and manipulate logarithmic expressions with powers.