Problem 15 - Algebra and Trigonometry with Analytic Geometry

Question

Write the expression as one logarithm. $$ \ln y^{3}+\frac{1}{3} \ln \left(x^{3} y^{6}\right)-5 \ln y $$

Step-by-Step Solution

Answer

The expression simplifies to \( \ln x \).

Step 1: Apply Power Rule for Logarithms

The power rule for logarithms states that \( a \ln b = \ln b^a \). In the expression, \( rac{1}{3} \ln (x^3y^6) \) can be rewritten using this rule. Calculate:\[ \frac{1}{3} \ln (x^3y^6) = \ln (x^3y^6)^{1/3} = \ln (x^{1}y^{2}) \]

Step 2: Apply Product Rule for Logarithms

The product rule for logarithms states that \( \ln a + \ln b = \ln(ab) \). Use this rule to combine \( \ln y^{3} \) and \( \ln (x^{1}y^{2}) \):\[ \ln y^{3} + \ln (x^{1}y^{2}) = \ln(y^{3} imes x^{1}y^{2}) = \ln(xy^{5}) \]

Step 3: Apply Quotient Rule for Logarithms

The quotient rule for logarithms states that \( \ln a - \ln b = \ln(\frac{a}{b}) \). Now, combine the resulting logarithm with \( -5 \ln y \) using the quotient rule:\[ \ln(xy^{5}) - 5\ln y = \ln \left( \frac{xy^{5}}{y^{5}} \right) = \ln x \]

Step 4: Final Expression

After applying these rules, the expression is simplified to one logarithm:\[ \ln x \]

Key Concepts

Power Rule of Logarithms

The power rule of logarithms is a fundamental concept that simplifies expressions involving logarithms with exponents. The rule states that if you have a logarithm of a number raised to a power, you can multiply the power by the logarithm of that number. Mathematically, this is expressed as:

• \( \ln(b^a) = a \ln(b) \)

This rule can transform complex expressions into simpler ones, easing the computation process.
For example, in our original problem, we applied the power rule to \( \frac{1}{3} \ln(x^3y^6) \), converting it to \( \ln(xy^2) \).
This transformation allows for easier manipulation and combination of logarithmic terms, preparing them for further simplification using other logarithmic properties.

Product Rule of Logarithms

The product rule of logarithms helps in combining logarithms of multiplied terms. According to this rule, the sum of logarithms is equivalent to the logarithm of the product of the individual terms:

• \( \ln(a) + \ln(b) = \ln(ab) \)

This rule simplifies expressions where multiple logarithms can be combined into a single one. In our exercise, we used the product rule when combining \( \ln y^3 \) and \( \ln (xy^2) \).
After applying the rule, these terms were expressed as \( \ln(xy^5) \).
By reducing the number of logarithmic terms, the problem becomes much more manageable and closer to achieving the final simplified expression.

Quotient Rule of Logarithms

The quotient rule of logarithms is applied when you need to simplify the difference between two logarithms. This rule states that the difference of logarithms is the logarithm of the quotient of the terms:

• \( \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) \)

This rule is essential in reducing expressions involving subtraction of logs into a single logarithm. In the given problem, this rule comes into play in the final step.
It combines \( \ln(xy^5) \) with \(-5 \ln y \), resulting in \( \ln \left(\frac{xy^5}{y^5}\right) \).
After cancellation, the expression simplifies to \( \ln x \), showing how efficient use of logarithmic properties can lead to a concise solution.