Problem 76 - Elementary and Intermediate Algebra

Question

Write logarithmic expression as one logarithm. \(2 \log x+\frac{1}{2} \log y\)

Step-by-Step Solution

Answer

\( \log(x^2 \sqrt{y}) \)

Step 1: Understand the Logarithm Identities

One of the main ideas here is to use the properties of logarithms to combine expressions. We will be using the power rule of logarithms: \( a \log b = \log b^a \), which allows us to move the coefficient of the logarithm inside as an exponent.

Step 2: Apply the Power Rule on Each Term

Apply the power rule to each term in the expression. \( 2 \log x \) becomes \( \log(x^2) \), and \( \frac{1}{2} \log y \) becomes \( \log(y^{1/2}) \).

Step 3: Combine Using the Product Rule

After rewriting the expression using the power rule, we use the product rule of logarithms to combine them: \( \log A + \log B = \log (A \times B) \). Thus, \( \log(x^2) + \log(y^{1/2}) \) can be combined into one logarithmic expression as \( \log(x^2 \cdot y^{1/2}) \).

Step 4: Final Expression Simplification

Combine and simplify the expression if necessary: \( \log(x^2 \cdot y^{1/2}) = \log(x^2 \sqrt{y}) \). This is the final expression written as one logarithm.

Key Concepts

Logarithmic Expressions

Logarithmic expressions can seem complex at first, but they are just a way to express and manipulate logarithms in various forms. In mathematics, a logarithm answers the question: "To what power must we raise the base, usually 10 or e, to get a specific number?" For example, the expression \(\log_b(a)\) asks which power, when raised on base \(b\), gives us the number \(a\). When you have multiple logarithmic expressions, you can use logarithm rules to combine them into simpler forms.

• Breaking Down the Terms: Each term in a logarithmic expression can be treated as a standalone part. For example, \( 2 \log x \) and \( \frac{1}{2} \log y \) are two separate parts that can be simplified further using logarithm rules.
• Co-efficients in Logarithms: Co-efficients in front of \( \log \) indicate how many times a logarithm is multiplied. For example, the '2' in \( 2 \log x \) is a co-efficient that can be moved inside the logarithm using rules.

Power Rule of Logarithms

The power rule of logarithms is an essential tool that helps you handle coefficients in logarithmic expressions. It states that any coefficient in front of a logarithm can be moved as an exponent inside the logarithm. The formula is expressed as \( a \log b = \log b^a \). This rule essentially shifts the multiplication outside of the logarithm into an exponent, making expressions easier to combine or simplify.

• Application of the Power Rule: Given \( 2 \log x \), you can rewrite it as \( \log(x^2) \). Similarly, \( \frac{1}{2} \log y \) becomes \( \log(y^{1/2}) \).
• Why Use It?: Applying the power rule helps in incorporating multiple terms into a single expression and is a stepping stone for using other logarithmic rules effectively.

Understanding and applying the power rule allows you to convert complicated terms into a simpler exponents form inside a logarithmic function.

Product Rule of Logarithms

Once you have applied the power rule of logarithms, you can move on to the product rule to combine the expressions. The product rule states that the logarithm of a product is the sum of the logarithms of the factors. In mathematical terms, it is expressed as \( \log A + \log B = \log (A \times B) \). This rule is incredibly useful in consolidating multiple logarithmic terms into one simplified expression.

• Combining Logs: After applying the power rule and getting expressions like \( \log(x^2) + \log(y^{1/2}) \), you can use the product rule to combine them into \( \log(x^2 \cdot y^{1/2}) \).
• Why It Matters: Using the product rule helps you streamline expressions, making them easier to work with, especially in both algebraic manipulations and solving equations.

Using the product rule, you will be able to get a final, simple form of a logarithmic expression, such as \( \log(x^2 \sqrt{y}) \), from more complex expressions.