Problem 30 - College Algebra Essentials

Question

Use properties of logarithms to expand logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. \(\ln \sqrt{e x}\)

Step-by-Step Solution

Answer

The expanded logarithmic expression is \(1/2 \cdot \ln x\).

Step 1: Rewrite Square Root as Exponent

Express the square root as an exponent. This gives us \(\ln ( (e x)^{1/2})\).

Step 2: Applying Logarithm Rule

By using logarithm rules, the exponent can be brought to the front of the logarithm. This results in \(1/2 \cdot \ln (e x)\).

Step 3: Simplify the Equation

Knowing that \(\ln e = 1\), we can simplify the equation further. This gives us \(1/2 \cdot \ln x\).

Key Concepts

Properties of Logarithms

Logarithms are powerful mathematical tools that simplify complex expressions and solve exponential equations. They have specific properties that allow us to manipulate expressions in useful ways. Two major properties include:

• Product Property: This states that the logarithm of a product equals the sum of the logarithms of the factors. Mathematically written as: \( \log_b (mn) = \log_b m + \log_b n \).
• Power Property: This allows you to bring down the exponent as a multiplier out front: \( \log_b (m^n) = n \cdot \log_b m \).

Using these properties simplifies the calculations and expands expressions into manageable chunks. They are key when dealing with complex expressions like \( \ln \sqrt{e x} \).

Expansion of Logarithmic Functions

When expanding a logarithmic expression, you break it into simpler parts by using the properties of logarithms. The goal is to express the logarithm in a sum or difference of simpler logarithms. For the expression \( \ln \sqrt{e x} \), you follow these steps:

• Convert the square root to an exponent: \( \sqrt{e x} = (e x)^{1/2} \).
• Apply the power property to bring the exponent in front: \( \ln((e x)^{1/2}) = \frac{1}{2} \cdot \ln(e x) \).
• If possible, further simplify by identifying known logarithmic values, such as \( \ln e = 1 \).

The expansion of logarithmic expressions reduces complexity and reveals finer details within the expression.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a logarithm with the base of the mathematical constant \( e \), approximately equal to 2.718. It is extensively used in mathematics owing to its natural properties related to calculus and growth rates.

• Key Identity: The natural logarithm of \( e \) is 1, written as: \( \ln e = 1 \). This simple identity is crucial for simplifying expressions.
• Applications: Natural logarithms are used in calculus, particularly in problems involving growth processes and integrals.

Given the exercise \( \ln \sqrt{e x} \), the property \( \ln e = 1 \) helps simplify \( \ln(e x) \) to \( \ln x \) after using the power property.

Logarithm Rules

To work with logarithmic expressions effectively, you need to remember the key rules of logarithms:

• Quotient Rule: For dividing within a logarithm, it breaks down to the difference: \( \log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n \).
• Power Rule: Bring down the exponent as a coefficient: \( \log_b (m^n) = n \cdot \log_b m \). This rule simplifies expressions greatly, as seen in \( \ln ((e x)^{1/2}) \).

Recognizing and applying these rules is essential when decomposing or combining logarithmic expressions. They enable mathematicians and students alike to handle more complex algebraic problems, providing solutions with elegance and simplicity.