Problem 38 - Precalculus

Question

Evaluate the expression. \(\ln \left(\frac{1}{\sqrt{e}}\right)\)

Step-by-Step Solution

Answer

The evaluated expression is \(-\frac{1}{2}\).

Step 1: Rewrite the Fraction

The expression inside the natural logarithm is a fraction: \( \frac{1}{\sqrt{e}} \). This can be rewritten using the exponent rule for square roots: \( \frac{1}{e^{1/2}} \).

Step 2: Apply Properties of Exponents

Employ the property of exponents: \( a^{-b} = \frac{1}{a^b} \). This changes our expression to \( e^{-1/2} \).

Step 3: Use the Natural Logarithm Property

Use the logarithm property \( \ln(a^b) = b \cdot \ln(a) \). In our case, \( \ln(e^{-1/2}) = -\frac{1}{2} \cdot \ln(e) \).

Step 4: Evaluate the Logarithm

Recall that the natural logarithm of \( e \), or \( \ln(e) \), is equal to 1. Therefore, \( -\frac{1}{2} \cdot \ln(e) = -\frac{1}{2} \cdot 1 \).

Step 5: Simplify the Expression

Multiply to complete the simplification: \( -\frac{1}{2} \cdot 1 = -\frac{1}{2} \). Therefore, the evaluated expression is \( -\frac{1}{2} \).

Key Concepts

Exponent Rules

Exponent rules are fundamental for dealing with powers in mathematics. They help to simplify expressions involving exponents. When you see an expression like \( \frac{1}{\sqrt{e}} \), the exponent rule allows you to rewrite it into a more manageable form. A square root can be expressed as a power of one-half, so \( \sqrt{e} \) is equivalent to \( e^{1/2} \).

Using the law \( a^{-b} = \frac{1}{a^b} \), you can convert \( \frac{1}{\sqrt{e}} \) into \( e^{-1/2} \).
This conversion simplifies mathematical operations as it enables using other exponent properties easily.

Remember, some basic exponent rules are:

• \( a^m \times a^n = a^{m+n} \)
• \( \frac{a^m}{a^n} = a^{m-n} \)
• \((a^m)^n = a^{m \cdot n}\)
• \(a^0 = 1\) for any \(aeq0\)

By mastering these rules, you can approach complex expressions with confidence.

Properties of Logarithms

The properties of logarithms provide powerful techniques to simplify expressions involving logs. Logarithms allow us to work with exponents as multiplication: \[ \ln(a^b) = b \cdot \ln(a) \]When working with the natural logarithm, \( \ln \), these properties become more pronounced as they let you pull down the exponent.
This process transforms products into sums, dividends into subtractions, and powers into multipliers.

For example, if you have \( \ln(e^{-1/2}) \), applying the property yields
\(-\frac{1}{2} \cdot \ln(e) \). This transformation simplifies logarithmic expressions greatly.

Some key properties to remember include:

• \(\ln(a \cdot b) = \ln(a) + \ln(b)\)
• \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\)
• \(\ln(a^b) = b \cdot \ln(a)\)

These properties unify the rules for handling exponential parts, thus making complex calculations much simpler.

Simplifying Expressions

Simplifying expressions involves breaking down and rewriting mathematical terms to their simplest forms. In our problem \( \ln\left(\frac{1}{\sqrt{e}}\right) \), each step helps us move closer to a simplified result.

First, by using exponent rules, we turn complicated roots and fractions into simpler exponents. This yields a straightforward expression that is easier to handle.
Then, applying the properties of logarithms, we transform exponents into multipliers, making the expression clearer and easier to evaluate.

Finally, it's vital to remember basic logarithmic values such as \( \ln(e) = 1 \), since these shortcuts dramatically simplify evaluations.
If you consider our problem, after simplification through properties and rules, we find that \(-\frac{1}{2} \cdot \ln(e)\) becomes \(-\frac{1}{2} \).
This showcases how expression simplification combines concepts to find easy, elegant solutions to complex problems.

Approaching mathematical expressions with these methods can streamline much of the common confusion involved, making math not only easier but more enjoyable.