Question

Expand the logarithmic expression. (See Example 2.) $$\ln \frac{6 x^2}{y^4}$$

Short answer

The expanded form of \( \ln \frac{6 x^2}{y^4} \) is \( \ln 6 + 2\ln x - 4\ln y \)

Step-by-step solution

Apply the Quotient Rule

Using the quotient rule of logarithms which states that \( \ln a/b = \ln a - \ln b \), the given expression can be rewritten as: \( \ln 6x^2 - \ln y^4 \)

Apply the Product Rule

Then, apply the product rule of logarithms. This rule states that the \( \ln ab = \ln a + \ln b \). So, \( \ln 6x^2 \) can be rewritten as \( \ln 6 + \ln x^2 \)

Apply the Power Rule

Finally, apply the power rule to both \( \ln x^2 \) and \( \ln y^4 \). The power rule of logarithms states that \( \ln a^n = n \ln a \). Hence, \( \ln x^2 \) becomes \( 2\ln x \) and \( \ln y^4 \) becomes \( 4\ln y \)

Write the Final Answer

Now, combine all the steps. Therefore, the expanded form of \( \ln \frac{6 x^2}{y^4} \) is \( \ln 6 + 2\ln x - 4\ln y \)

Key concepts

Quotient Rule

The quotient rule for logarithms is a handy tool that helps simplify logarithmic expressions involving divisions. When you have a logarithm of a fraction, such as \( \ln \frac{a}{b} \), you can split it into two separate logarithms. This rule states that the logarithm of a quotient is equal to the logarithm of the numerator minus the logarithm of the denominator.

Using this rule can make complex expressions more manageable. In the original exercise, we started with \( \ln \frac{6x^2}{y^4} \). By applying the quotient rule, it becomes \( \ln 6x^2 - \ln y^4 \). This step sets the stage for further simplifying the expression using other logarithmic rules.

Remember:

• The logarithm of a quotient: \( \ln \frac{a}{b} = \ln a - \ln b \).
• Each term should be simplified using other rules after applying the quotient rule.

Product Rule

The product rule is another essential tool for dealing with logarithms. It comes into play when you're working with the logarithm of a product, such as \( \ln(ab) \). The rule tells us that this is equal to the sum of the logarithms of the individual factors. So, \( \ln(ab) = \ln a + \ln b \).

In our exercise, after we applied the quotient rule, we used the product rule on \( \ln 6x^2 \). By doing this, the expression splits into \( \ln 6 + \ln x^2 \). The product rule helps break down products into separate parts, making it easier to apply further rules like the power rule.

Key points about the product rule:

• The logarithm of a product: \( \ln(ab) = \ln a + \ln b \).
• This step is crucial when factors within a term can still be further simplified using additional rules.

Power Rule

The power rule is instrumental when you have exponents within logarithmic expressions. It states that the logarithm of an exponential term can be rewritten by bringing down the exponent in front as a multiplier. For instance, \( \ln(a^n) = n \ln a \).

In the step-by-step solution provided, once the product and quotient rules were applied, the power rule was used to simplify \( \ln x^2 \) to \( 2\ln x \), and \( \ln y^4 \) to \( 4\ln y \). This rule efficiently transforms expressions involving powers into something more manageable.

Insights on the power rule include:

• Exponential logarithms transform by moving the exponent: \( \ln(a^n) = n \ln a \).
• Allows for clear and easy simplification of terms that were once complex due to exponents.