Problem 13 - Algebra and Trigonometry with Analytic Geometry

Question

Write the expression as one logarithm. $$ \log \left(x^{3} y^{2}\right)-2 \log x \sqrt[3]{y}-3 \log \left(\frac{x}{y}\right) $$

Step-by-Step Solution

Answer

\( \log \left( \frac{y^{13/3}}{x^2} \right) \)

Step 1: Apply the Power Rule of Logarithms

Recall that the power rule of logarithms states that \( a \log b = \log b^a \). Apply this rule to each term in the expression:\[2\log x \sqrt[3]{y} = \log (x\sqrt[3]{y})^2 = \log (x^2 y^{2/3}) \]\[3 \log \left(\frac{x}{y}\right) = \log \left(\frac{x}{y}\right)^3 = \log \left(\frac{x^3}{y^3}\right) \]The expression now becomes:\[\log (x^3 y^2) - \log (x^2 y^{2/3}) - \log \left(\frac{x^3}{y^3}\right)\]

Step 2: Apply the Quotient Rule of Logarithms

The quotient rule of logarithms states that \( \log a - \log b = \log \frac{a}{b} \). Apply this rule to the modified expression:\[ \log \frac{x^3 y^2}{x^2 y^{2/3}} - \log \left(\frac{x^3}{y^3}\right) \]

Step 3: Simplify the Expression

Simplify the expression within each logarithm by performing division and applying the laws of exponents:\[ \log \frac{x^3 y^2}{x^2 y^{2/3}} = \log \left( x^{3-2} y^{2-(2/3)} \right) = \log \left( x^1 y^{4/3} \right) \]Now substitute back into the expression:\[ \log (x y^{4/3}) - \log \left(\frac{x^3}{y^3}\right) \]

Step 4: Combine the Logarithms Using the Quotient Rule Again

Combine the two logarithms using the quotient rule once more:\[ \log \frac{x y^{4/3}}{\frac{x^3}{y^3}} \]Perform division:\[ \frac{x y^{4/3} \cdot y^3}{x^3} = \frac{y^{4/3 + 3}}{x^{3-1}} = \frac{y^{13/3}}{x^2} \]

Step 5: Final Step: Express as One Logarithm

The final expression simplifies to:\[ \log \left( \frac{y^{13/3}}{x^2} \right) \]

Key Concepts

Power Rule of Logarithms

The Power Rule of Logarithms is a fundamental tool in simplifying complex logarithmic expressions. This rule states that if you have an expression of the form \( a \log b \), you can rewrite it as \( \log b^a \). This is especially useful when dealing with logarithms involving exponents.

Imagine you have \( 2\log x \sqrt[3]{y} \). Using the power rule, this can be changed into \( \log (x \sqrt[3]{y})^2 \). This becomes \( \log (x^2 y^{2/3}) \) by applying the power inside the logarithm to both \( x \) and \( y \).

• It simplifies expressions by combining multiplication and exponentiation under a single logarithm.
• Keeps calculations tidy by reducing the number of operations you have to perform.

With this rule, you're able to tame otherwise unwieldy logarithmic terms by transforming them into a more manageable format.

Quotient Rule of Logarithms

The Quotient Rule of Logarithms is a powerful mechanism for handling subtraction between logarithmic expressions. It is expressed by the formula \( \log a - \log b = \log \frac{a}{b} \). This allows a subtraction of logs to be combined into a single log.

In practice, consider the expression \( \log (x^3 y^2) - \log (x^2 y^{2/3}) \). We can apply the quotient rule to write it as \( \log \frac{x^3 y^2}{x^2 y^{2/3}} \).

• Useful for simplifying expressions, making them more compact.
• Transforms complex subtractive relationships in logs into division inside a single logarithm.

Utilizing this rule ensures that you can navigate logarithmic expressions with multiple components efficiently.

Laws of Exponents

The Laws of Exponents play a crucial role when handling expressions inside a logarithm, especially during simplification. These laws govern how we combine and manipulate powers of the same base.

In the context of logarithms, let's say you have \( \frac{x^3 y^2}{x^2 y^{2/3}} \). Here, you need to apply the laws such as:

• When dividing like bases, you subtract their exponents: \( x^{3-2} = x^1 \).
• When dealing with \( y \), \( y^{2-(2/3)} = y^{4/3} \).

By using these laws, the base elements inside the fractional expression are simplified, resulting in fewer terms which makes further operations and understanding more straightforward.

Simplifying Logarithms

Simplifying logarithmic expressions involves applying rules systematically until you reach the simplest form possible. This means combining rules of logarithms and exponents to condense everything into a single, clear expression.

Take for instance transforming \( \log \frac{x y^{4/3}}{\frac{x^3}{y^3}} \) into a simplified form. By performing careful division, you manage to reorganize it to \( \frac{y^{13/3}}{x^2} \).

• Process includes employing exponent and log rules strategically.
• Ensures maximum reduction of expression complexity making further mathematical manipulation easier.

The simplification process always aims to express your result in the most compact form, minimizing cognitive load when analyzing or solving problems.