Problem 60 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.)$$\ln \sqrt{\frac{x^{2}}{y^{3}}}$$.

Step-by-Step Solution

Answer

The expanded form of \(\ln \sqrt{\frac{x^{2}}{y^{3}}}\) is \(\ln x - 1.5 \ln y\).

Step 1: Rewrite the Square Root as an Exponent

The square root of a quantity can be represented as a power of one-half according to the rules of exponents. Therefore, rewrite the \(\ln \sqrt{\frac{x^{2}}{y^{3}}}\) as \(\ln (\frac{x^{2}}{y^{3}})^{0.5}\).

Step 2: Use Power Rule of Logarithms

The power rule of logarithms states that \(\ln a^b = b \ln a\). When you apply the power rule, the initial expression \(\ln (\frac{x^{2}}{y^{3}})^{0.5}\) becomes \(0.5\ln \frac{x^{2}}{y^{3}}\).

Step 3: Apply Quotient Rule of Logarithms

The quotient rule of logarithms states that \(\ln \frac{a}{b} = \ln a - \ln b\). When you apply this rule, it transforms our latest expression \(0.5\ln \frac{x^{2}}{y^{3}}\) into \(0.5(\ln x^{2} - \ln y^{3})\).

Step 4: Use Power Rule Again

Now let’s apply the power rule of logarithms again to simplify the expression. Each of the expressions within the parenthesis separately according to the power rule, \(\ln a^{n} = n \ln a\), which results in \(0.5(2\ln x - 3\ln y)\).

Step 5: Distribute Multiplication Over Parentheses

The next step is to distribute the 0.5 (which is the exponent from the original square root transformation) inside the parenthesis. The expression becomes: \(0.5*2\ln x - 0.5*3\ln y = \ln x - 1.5 \ln y\).

Key Concepts

Power Rule

The power rule is an essential property in logarithms that simplifies the process of dealing with exponents within a logarithmic expression. According to the power rule, when you have a logarithmic expression of the form \( \ln(a^b) \), you can bring the exponent \( b \) in front of the logarithm as a multiplier: \( b \ln(a) \).
This is extremely useful when you want to break down complex expressions.

• Example: \( \ln(x^4) \) can be rewritten as \( 4\ln(x) \).
• This rule allows us to deal with exponents more conveniently, making it possible to handle large powers more easily.

In the exercise, the power rule was applied twice. First, it converted \( \ln \left(\frac{x^{2}}{y^{3}}\right)^{0.5} \) into \( 0.5 \ln \left(\frac{x^{2}}{y^{3}}\right) \).
Then, it further simplified the expression to \( 0.5(2\ln x - 3\ln y) \). Each term within the parentheses utilized the power rule separately.

Quotient Rule

The quotient rule is another helpful logarithmic property used when you need to simplify logarithms involving fractions. The quotient rule states that the logarithm of a fraction is the difference between the logarithm of the numerator and the denominator: \( \ln \left(\frac{a}{b}\right) = \ln(a) - \ln(b) \).

• This allows you to express the logarithm of a division as a subtraction, simplifying the expression significantly.
• Example: \( \ln \left(\frac{x}{y}\right) \) becomes \( \ln(x) - \ln(y) \).

In the given exercise, the quotient rule was crucial in handling the fraction inside the logarithm when the step progressed from \( 0.5 \ln \left(\frac{x^{2}}{y^{3}}\right) \) to \( 0.5 (\ln x^{2} - \ln y^{3}) \).
This step clearly illustrates how dividing the expression into smaller, more manageable parts can make the logarithmic operation straightforward.

Expanding Logarithms

Expanding logarithms involves breaking down complex logarithmic expressions into a sum, difference, or multiple of simpler logarithms by leveraging the properties like the power and quotient rules.
Expanding is particularly useful for simplifying expressions to make them easier to understand or compute.

• Start by applying the power rule to handle any exponents.
• Use the quotient rule to separate fractions into individual terms.
• Finally, simplify each term such that individual powers and coefficients are clear.

For the given exercise, expanding the logarithm \( \ln \sqrt{\frac{x^{2}}{y^{3}}} \) was accomplished by rewriting it accurately with the steps: first, converting the square root to an exponent, then applying the quotient rule, and using the power rule to fully expand it to \( \ln x - 1.5 \ln y \).
This method helps break down complex expressions into more comprehensible parts, fitting perfectly within the toolbox of logarithmic properties to simplify your calculations.