Problem 14 - Algebra and Trigonometry with Analytic Geometry

Question

Write the expression as one logarithm. $$ 2 \log \frac{y^{3}}{x}-3 \log y+\frac{1}{2} \log x^{4} y^{2} $$

Step-by-Step Solution

Answer

Combine and simplify to \( \log(y^{10}) \).

Step 1: Apply Logarithm Power Rule

The logarithm power rule states that \( a \log b = \log b^a \). Apply this rule to each term: - For the first term: \( 2 \log \frac{y^3}{x} \) becomes \( \log\left(\frac{y^3}{x}\right)^2 \).- For the second term: \(-3 \log y \) becomes \( \log y^{-3} \).- For the third term: \( \frac{1}{2} \log x^4 y^2 \) becomes \( \log(x^4 y^2)^{1/2} \).

Step 2: Simplify Each Logarithmic Expression

Simplify each logarithmic expression:- \( \log\left(\frac{y^3}{x}\right)^2 \) simplifies to \( \log\left(\frac{y^6}{x^2}\right) \).- \( \log(y^{-3}) \) is already simplified.- \( \log(x^4 y^2)^{1/2} \) simplifies to \( \log(x^2 y) \).

Step 3: Use Logarithm Addition and Subtraction Rules

Combine the three simplified logarithmic expressions using logarithm rules:- Start with \( \log\left(\frac{y^6}{x^2}\right) - \log(y^{-3}) + \log(x^2 y) \).- The subtraction of logs becomes a division: \( \log\left(\frac{\frac{y^6}{x^2}}{y^{-3}}\right) + \log(x^2 y) \).- Using the addition of logs, this converts to: \( \log\left(\frac{\frac{y^6}{x^2}}{y^{-3}} \cdot x^2 y \right) \).

Step 4: Simplify the Combined Logarithmic Expression

Simplify the expression from the previous step:- Rewrite \( \frac{\frac{y^6}{x^2}}{y^{-3}} \) as \( \frac{y^{6+3}}{x^2} = \frac{y^9}{x^2} \).- Multiply by \( x^2 y \) to get \( \frac{y^9}{x^2} \cdot x^2 y = y^9 y = y^{10} \).- So the expression simplifies to \( \log(y^{10}) \).

Step 5: Final Simplified Expression

The final expression \( \log(y^{10}) \) can be simplified to \( 10 \log y \) using the power rule, if needed. However, as a single logarithm, it's best expressed as \( \log(y^{10}) \).

Key Concepts

Logarithm Power Rule

The logarithm power rule is an essential tool in simplifying logarithmic expressions. This rule allows us to "move" the coefficient in front of a logarithm inside as an exponent. Put simply, for any logarithmic expression like \( a \log b \), you can convert it to \( \log b^a \).
This transformation helps in rewriting logarithms in their simplest form, making it easier to solve more complex equations later on.

• For example, the expression \( 2 \log \frac{y^3}{x} \) was rewritten as \( \log\left(\frac{y^3}{x}\right)^2 \), which simplifies the multiplication factor to an exponent.
• The term \(-3 \log y\) becomes \(\log y^{-3}\), making it easier to handle in calculations.

Next time you face such expressions, remember this rule to simplify calculations.

Logarithm Addition and Subtraction Rules

Once each term has been simplified using the power rule, the next step involves combining these simplified expressions. This is where the logarithm addition and subtraction rules come into play. These rules can transform complex expressions into more manageable forms.
The rules are:

• When you add two logarithms with the same base, you can transform the expression into a single logarithm of the product: \( \log a + \log b = \log(ab) \).
• When you subtract one logarithm from another with the same base, you turn the expression into a single logarithm of the quotient: \( \log a - \log b = \log\left(\frac{a}{b}\right) \).

In our exercise, for instance, the subtraction step \( \log\left(\frac{y^6}{x^2}\right) - \log(y^{-3}) \) was combined into \( \log\left(\frac{\frac{y^6}{x^2}}{y^{-3}}\right) \), making it easier to simplify further. Use these rules to smartly manage and simplify logarithmic equations.

Simplification of Logarithmic Expressions

After applying the power rule and combining terms using addition and subtraction rules, the final step is to simplify these expressions as much as possible. Simplification often means writing the expression in its most condensed form.
In the given problem, this involved:

• Rewriting terms that contain subtractions as divisions and subtractions as multiplications.
• The expression \( \frac{\frac{y^6}{x^2}}{y^{-3}} \) was simplified to \( \frac{y^{6+3}}{x^2} = \frac{y^9}{x^2} \).
• Combining expressions using multiplication such as \( \frac{y^9}{x^2} \cdot x^2 y \), which eventually gave \( y^{10} \).

As a final result, a single logarithm \( \log(y^{10}) \) is obtained. Remember, simplification is about reducing complexity while maintaining equivalency.
Mastering these simplification techniques will greatly help in tackling logarithmic expressions in more advanced topics.