Problem 82 - Algebra 2

Question

Expand each logarithm. \(\log _{b} \frac{\sqrt{x} \sqrt[3]{y^{2}}}{\sqrt[5]{z^{2}}}\)

Step-by-Step Solution

Answer

The expanded form of the given logarithm is \(\frac{1}{2}\log_{b}{x} + \frac{2}{3}\log_{b}{y} - \frac{2}{5}\log_{b}{z}\)

Step 1: Apply the Quotient Rule

The quotient rule of logarithms states that the logarithm of a quotient is the difference of the logarithms of the numerator and the denominator. Therefore, \(\log _{b} \frac{\sqrt{x} \sqrt[3]{y^{2}}}{\sqrt[5]{z^{2}}}\) can be rewritten as \(\log_{b}{(\sqrt{x} \sqrt[3]{y^{2}})} - \log_{b}{(\sqrt[5]{z^{2}})}\).

Step 2: Apply the Product Rule

The product rule of logarithms converts the multiplication of terms inside a log into addition outside the logarithm. Apply it to \(\log_{b}{(\sqrt{x} \sqrt[3]{y^{2}})}\) portion of the equation, we will get \(\log_{b}{\sqrt{x}} + \log_{b}{\sqrt[3]{y^{2}}}\). So, now our equation becomes \((\log_{b}{\sqrt{x}} + \log_{b}{\sqrt[3]{y^{2}}}) - \log_{b}{\sqrt[5]{z^{2}}}\)

Step 3: Apply the Power Rule

The power rule allows the exponent of the argument of a logarithm to be brought out as a multiplier. Apply this rule to every term of the equation. We get \(\frac{1}{2}\log_{b}{x} + \frac{2}{3}\log_{b}{y} - \frac{2}{5}\log_{b}{z}\).

Step 4: Conclusion

Finally, the expanded form of the given logarithm is \(\frac{1}{2}\log_{b}{x} + \frac{2}{3}\log_{b}{y} - \frac{2}{5}\log_{b}{z}\).

Key Concepts

Quotient Rule of Logarithms

The quotient rule of logarithms is a helpful tool when dealing with logarithms that involve division. According to this rule, the logarithm of a quotient can be expressed as the difference of two separate logarithms. For example, if you have the expression \(\log_{b} \left( \frac{A}{B} \right)\), it can be transformed into \(\log_{b}{A} - \log_{b}{B}\). This property makes it easier to work with complicated divisions inside a logarithm by breaking them down into simpler components.

In our given problem, this rule is applied to \(\log_{b} \frac{\sqrt{x} \sqrt[3]{y^{2}}}{\sqrt[5]{z^{2}}}\) to give us \(\log_{b}{(\sqrt{x} \sqrt[3]{y^{2}})} - \log_{b}{(\sqrt[5]{z^{2}})}\). This breakdown sets the stage for further simplification using other logarithmic rules.

Product Rule of Logarithms

The product rule of logarithms is key when you have a product inside a logarithm and want to simplify it. It states that the logarithm of a product can be turned into the sum of the logarithms of each individual factor. If you encounter \(\log_{b}(A \cdot B)\), it simplifies to \(\log_{b}{A} + \log_{b}{B}\). This rule helps make complex logarithmic expressions much more manageable.

In our problem, after using the quotient rule, we apply the product rule to the term \(\log_{b}{(\sqrt{x} \sqrt[3]{y^{2}})}\). This changes it to \(\log_{b}{\sqrt{x}} + \log_{b}{\sqrt[3]{y^{2}}}\). Breaking down the multiplication within the logarithm as an addition of separate logs, allows us to handle each part independently.

Power Rule of Logarithms

The power rule of logarithms is a powerful technique that helps in handling logarithms with exponents. This rule lets you change the exponent of the argument of the log into a coefficient in front of the log. Mathematically, it's expressed as \(\log_{b}{(A^n)} = n \cdot \log_{b}{A}\). This rule makes it easier to work with logarithms involving power terms and simplify expressions.

In our example, we utilize the power rule in the final step to each term resulting from the previous applications of the quotient and product rules. By applying the power rule, \(\log_{b}{\sqrt{x}}\) simplifies to \(\frac{1}{2}\log_{b}{x}\), \(\log_{b}{\sqrt[3]{y^{2}}}\) becomes \(\frac{2}{3}\log_{b}{y}\), and \(\log_{b}{\sqrt[5]{z^{2}}}\) simplifies to \(\frac{2}{5}\log_{b}{z}\). Each exponent inside the logarithmic terms is brought out as a fraction, making the expression far simpler.