Problem 51 - Algebra and Trigonometry

Question

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ \frac{1}{2} \ln x+\ln y $$

Step-by-Step Solution

Answer

\(\ln(x^{1/2}y)\)

Step 1: Utilize property of the logarithm

Start with the equation \(\frac{1}{2}\ln x+\ln y\). Apply the logarithmic property \(\ln(a^n) = n\ln(a)\) to rewrite \(\frac{1}{2}ln(x)\) as \(\ln(x^{1/2})\). This will give you \(\ln(x^{1/2}) + \ln(y)\)

Step 2: Combine logarithmic expressions

After rewriting the first component in step 1, you would have two logarithmic expressions added together: \(\ln(x^{1/2}) + \ln(y)\). So, you can apply the logarithmic property \(\ln(a) + \ln(b) = \ln(ab)\) to combine these two logarithmic expressions. This results in the expression being condensed into a single logarithm: \(\ln(x^{1/2}y)\)

Step 3: Simplify and finalize answer

The final expression after condensing the logarithms is \(\ln(x^{1/2}y)\). This is a simplified form of the original problem and cannot be simplified further. This is the final answer.

Key Concepts

Condensing Logarithms

Condensing logarithms involves combining multiple logarithmic terms into a single logarithm. This process simplifies expressions, making them easier to work with.
The key property used to condense logarithms is the Product Property, which states that the sum of logarithms with the same base can be written as a single logarithm of the product of the terms.

• If you have \(\ln(a) + \ln(b)\), it can be rewritten as \(\ln(ab)\).

To condense an expression like \(\frac{1}{2}\ln x + \ln y\), we first use another logarithmic property, the Power Rule. This allows \(\frac{1}{2}\ln x\) to be rewritten as \(\ln(x^{1/2})\), which is helpful because expressing exponents within the logarithm allows for easier combination of terms.
After adjusting each term, we apply the Product Property to merge them into one logarithmic term: \(\ln(x^{1/2}y)\).
This approach not only streamlines calculations but also provides a more unified view of the logarithmic expression.

Logarithmic Expressions

Logarithmic expressions involve variables and constants under the scope of the logarithm function, indicating powers to which a base number is raised to produce another number. In expressions like \(\ln(x^{1/2}) + \ln(y)\), each component inside the logarithm contains valuable information about multiplication or division due to their properties.
Understanding components of these expressions means knowing how different rules apply to manipulate terms:

• The Power Rule allows adjusting coefficients into exponents of the argument, simplifying partial components like turning \(\ln(x^{1/2})\) from \(\frac{1}{2}\ln x\).
• The Product Rule helps merge \(\ln(a) + \ln(b)\) forms into a single expression \(\ln(ab)\) for simplicity and efficiency.

Working with logarithmic expressions means appreciating how these rules simplify analysis, making sense of complex equations and assisting in applications across mathematics.

Simplifying Logarithms

Simplifying logarithms involves reducing expressions to their most concise form, using logarithmic rules and properties. This approach is essential for problem-solving, especially when dealing with complex equations.
The simplification process reduces redundant or complicated expressions, often involving powers, products, or even quotients of terms:

• Using the Power Rule transforms coefficients into exponents, streamlining the expression.
• Applying the Product Rule lets you combine multiple logarithms into one, such as turning \(\ln(x^{1/2}) + \ln(y)\) into \(\ln(x^{1/2}y)\).

This not only makes expressions easier to interpret but also supports computational efficiency when evaluating further operations. Simplified logarithms are more practical for mathematical modeling, solving equations, or even when performing calculus operations.
Overall, by transforming and rearranging terms using established rules and properties, simplified logarithms reflect a cleaner, more elegant formulation of the original expression, offering clearer insight and easier manipulation.