Question

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

Short answer

The expanded form of the given expression is \(\ln{x} - \frac{3}{2} \ln{y}\)

Step-by-step solution

Apply the square root property

Recall that the square root is equivalent to an exponent of \(\frac{1}{2}\). Thus, \(\ln \sqrt{\frac{x^{2}}{y^{3}}}\) can also be written as \(\ln \big(\frac{x^{2}}{y^{3}}\big)^{\frac{1}{2}}\).

Apply the power rule

The power of a logarithm can be pulled out front as a coefficient. This gives \(\frac{1}{2} \ln \big(\frac{x^{2}}{y^{3}}\big)\).

Applying the quotient rule

The quotient inside a logarithm becomes subtraction between the logarithms. We get \(\frac{1}{2} \big(\ln x^{2} - \ln y^{3}\big)\).

Apply the power rule again

Similar to step 2, we can pull the power out of the logarithm, resulting in \(\frac{1}{2} \big(2\ln x - 3\ln y\big)\). Simplifying gives \(\ln{x} - \frac{3}{2} \ln{y}\).

Key concepts

Logarithm Expansion

Understanding logarithm expansion is essential for simplifying complex logarithmic expressions. Logarithm expansion utilizes properties of logarithms to rewrite a single logarithmic statement into a series of simpler terms that can be more easily interpreted or solved. For instance, the original exercise given shows the logarithm of a square root, which can be intimidating at first glance. However, by viewing the square root as an exponent, \( \sqrt{\frac{x^{2}}{y^{3}}} \) becomes \( \left(\frac{x^{2}}{y^{3}}\right)^{\frac{1}{2}} \) because the square root of any variable is the same as raising that variable to the \( \frac{1}{2} \) power. This is a crucial step in expanding logarithms because it transforms the problem into a form where other logarithmic properties can be applied.

Expanding logarithms allows us to move from an equation that may seem daunting to a set of operations involving individual logarithms of the numerator and denominator, or logarithms of values raised to a power. This not only makes them easier to handle algebraically but also lays the groundwork for further manipulation using rules like the logarithmic power and quotient rules.

Logarithmic Power Rule

The logarithmic power rule is a game-changer when dealing with expressions where a logarithm is applied to a base raised to an exponent. It states that \( \log_{b}(x^n) = n \log_{b}(x) \), effectively allowing us to 'pull out' the exponent out front as a multiplier of the logarithm. With reference to the exercise, \( \ln(\frac{x^{2}}{y^{3}}) \) can have its numerator and denominator dealt with separately according to the logarithmic power rule.

This is illustrated in Step 2 where the exponent \( \frac{1}{2} \) is brought out in front, and again in Step 4 where the same rule applies to \( \ln x^{2} \) and \( \ln y^{3} \), transforming them into \( 2\ln x \) and \( 3\ln y \) respectively. Understanding when and how to apply the logarithmic power rule is critical as it simplifies expressions and paves the way for easier manipulation, which, in turn, allows one to solve logarithmic equations more efficiently.

Logarithmic Quotient Rule

The logarithmic quotient rule enables us to streamline logarithmic expressions that involve division within the logarithm. According to this rule, \( \log_{b}\left(\frac{x}{y}\right) = \log_{b}(x) - \log_{b}(y) \), transforming the log of a quotient into the difference between two logs. In the context of our original logarithmic expression, once we've processed the radical as an exponent, we're left with \( \ln \big(\frac{x^{2}}{y^{3}}\big)^{\frac{1}{2}} \), which the quotient rule can handle adeptly.

In Step 3, the rule is applied to \( \ln \big(\frac{x^{2}}{y^{3}}\big) \) resulting in \( \ln x^{2} - \ln y^{3} \). This step critically simplifies the logarithm of a fraction to a subtraction problem, breaking down the complex fraction into individual components that are far less daunting to work with. Mastering the logarithmic quotient rule is thus indispensable for students tackling logarithmic expressions in algebra, calculus, and beyond, providing a clear path through potentially challenging logarithmic terrain.