Problem 43 - Precalculus

Question

Write each expression as a single logarithm. $$2 \log u-3 \log v-2 \log z$$

Step-by-Step Solution

Answer

\( \log \left(\frac{u^2}{v^3z^2}\right) \)

Step 1: Apply Power Rule to Each Logarithm

The power rule of logarithms states that \( a \log b = \log b^a \). Using this rule, we can rewrite each logarithm in the expression:- For \( 2 \log u \), apply the power rule to get \( \log u^2 \).- For \( -3 \log v \), apply the power rule to get \( \log v^{-3} \).- For \( -2 \log z \), apply the power rule to get \( \log z^{-2} \).

Step 2: Combine Logarithms Using the Quotient and Product Rules

The log quotient rule states that \( \log a - \log b = \log \left(\frac{a}{b}\right) \) and the log product rule states that \( \log a + \log b = \log (ab) \). Combining all the logs we have:\[ \log u^2 - \log v^3 - \log z^2 = \log \left(\frac{u^2}{v^3z^2}\right) \]

Step 3: Verify the Simplified Single Logarithm

Ensure that each transformation and combination was applied correctly. All operations should respect the properties of logarithms. The final expression is \( \log \left(\frac{u^2}{v^3z^2}\right) \).

Key Concepts

Power Rule of Logarithms

The power rule of logarithms is a fundamental concept that allows us to simplify logarithmic expressions. This rule states that given an expression of the form \( a \log b \), it can be simplified to \( \log b^a \). In simpler terms, the coefficient in front of the logarithm can be moved as an exponent on the argument of the logarithm.
For example, if you encounter \( 2 \log u \), you can rewrite it as \( \log u^2 \). Similarly, \( -3 \log v \) can be rewritten as \( \log v^{-3} \), and \( -2 \log z \) as \( \log z^{-2} \). This transformation is particularly useful in making the next steps of combining logarithms more straightforward.
Remember, applying the power rule correctly can significantly simplify complex expressions, saving time and effort.

Quotient Rule of Logarithms

The quotient rule of logarithms is incredibly useful when dealing with the subtraction of two logarithmic expressions. According to this rule, \( \log a - \log b \) is equivalent to \( \log \left(\frac{a}{b}\right) \).
This means that whenever you see subtraction between two logarithms, you can combine them into a single logarithm by dividing the arguments. For example, if you have \( \log u^2 - \log v^3 \), you rewrite it as \( \log \left(\frac{u^2}{v^3}\right) \).

• It simplifies subtraction into division within the logarithm.
• It helps in reducing the number of logarithms in an expression.
• It makes complex logarithmic expressions more manageable.

Be cautious, as the order of terms is important; always ensure that subtraction is handled correctly to avoid errors.

Product Rule of Logarithms

When you need to combine logarithms using addition, the product rule of logarithms comes into play. This rule states that \( \log a + \log b \) equals to \( \log (ab) \). It's an excellent tool for compressing multiple logarithmic terms into a single, more manageable expression.
In our initial problem, combining \( \log u^2 \), \( \log v^{-3} \) and \( \log z^{-2} \) involves using both the product and quotient rules to arrive at a single logarithmic expression.

• The addition of logs leads to multiplication inside the log.
• It simplifies expressions by reducing the number of logarithmic terms.
• It clarifies complex relationships within a logarithmic context.

Correct application of the product rule, often in tandem with the quotient rule, is essential for simplifying more intricate logarithmic equations. Take your time with these transformations to ensure accuracy.