Problem 35 - Algebra 2 and Trigonometry

Question

Expand each expression using the properties of logarithms. \(\log _{e} \sqrt{x}\)

Step-by-Step Solution

Answer

The expanded form is \( \frac{1}{2} \cdot \log_e(x) \).

Step 1: Recognize the Problem Type

We need to expand the expression \( \log_e \sqrt{x} \) using the properties of logarithms. This involves rewriting the expression in a different form using known logarithmic identities.

Step 2: Apply the Exponent Rule

Recall that the square root of \( x \) can be expressed as \( x^{1/2} \). The logarithmic identity for the exponent rule is \( \log_b(a^n) = n \cdot \log_b(a) \).

Step 3: Rewrite the Expression

Substitute \( \sqrt{x} \) with \( x^{1/2} \) in the expression. So, \( \log_e \sqrt{x} = \log_e(x^{1/2}) \).

Step 4: Apply the Exponent Rule to the Logarithmic Expression

Using the identity \( \log_b(a^n) = n \cdot \log_b(a) \), rewrite the expression: \( \log_e(x^{1/2}) = \frac{1}{2} \cdot \log_e(x) \). This completes the expansion of the expression.

Key Concepts

Exponent Rule

Understanding the exponent rule in logarithms is crucial for expanding and manipulating logarithmic expressions. The exponent rule states that \[\log_b(a^n) = n \cdot \log_b(a)\]This means that if you have a logarithm of a power of a number, you can multiply the logarithm of the base by the exponent. In our exercise, we used this rule to transform the expression \( \log_e(\sqrt{x}) \). Since the square root of \( x \) is the same as \( x^{1/2} \), the expression becomes:\[\log_e(x^{1/2})\]Applying the exponent rule here, we move the exponent, \( 1/2 \), in front of the logarithm, resulting in:\[\frac{1}{2} \cdot \log_e(x)\]This makes the expression easier to work with and is a foundational step in the process of expansion.

Logarithmic Identities

Logarithmic identities are powerful tools for simplifying and transforming expressions involving logarithms. These identities help us understand how logarithms interact with algebraic expressions and provide ways to rewrite these expressions in a more manageable form. Some core identities include:

• Product Rule: \( \log_b(x \cdot y) = \log_b(x) + \log_b(y) \)
• Quotient Rule: \( \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \)
• Power Rule: \( \log_b(a^n) = n \cdot \log_b(a) \)

Using these identities, we can break down complex logarithmic expressions into simpler components. For instance, the problem we tackled involved recognizing \( \log_e(\sqrt{x}) \) as an application of the power rule, leading us to transform it into \( \frac{1}{2} \cdot \log_e(x) \). Each identity serves a specific purpose in expanding and simplifying logarithmic expressions.

Expansion of Logarithmic Expressions

Expanding logarithmic expressions is an essential skill in algebra and calculus. It involves using logarithmic identities to rewrite expressions in an expanded form, making complex problems more approachable. The expansion enables us to see the values and relationships within a logarithmic equation more clearly.In the problem of expanding \( \log _{e} \sqrt{x} \), we used the power rule to express the square root as a fractional exponent, giving us \( \log_e(x^{1/2}) \). Then, by applying the exponent rule, this expression expanded into:\[\frac{1}{2} \cdot \log_e(x)\]This simple form reveals how the logarithmic function handles roots and powers. Expansion is a powerful technique when solving equations or integrating complex logarithmic expressions, as it reduces the expression down to more fundamental components. Understanding this process is key to mastering logarithms in mathematics.