Problem 47 - Elementary and Intermediate Algebra

Question

Evaluate each expression without using a calculator. $$ \ln \sqrt[4]{e} $$

Step-by-Step Solution

Answer

The value of \(\ln \sqrt[4]{e}\) is \(\frac{1}{4}\).

Step 1: Identify the Expression Inside the Logarithm

The expression given is \(\ln \sqrt[4]{e}\). Notice that it involves a fourth root of \(e\), written as \(e^{1/4}\).

Step 2: Apply the Logarithmic Property

Recall the logarithmic identity \(\ln(a^b) = b \ln(a)\). We can rewrite the expression as \(\ln(e^{1/4})\).

Step 3: Simplify Using the Natural Logarithm Property

Knowing that \(\ln(e) = 1\), apply the identity from Step 2. Thus, \(\ln(e^{1/4}) = \frac{1}{4}\ln(e) = \frac{1}{4} \cdot 1 = \frac{1}{4}\).

Step 4: Final Simplification

There is no further simplification needed, so the expression evaluates to \(\frac{1}{4}\).

Key Concepts

Understanding Logarithmic Identities

Logarithmic identities are essential for simplifying logarithmic expressions and solving complex problems. A key identity to remember is the power rule for logarithms, which states: \( \ln(a^b) = b \ln(a) \).
This rule helps to take exponents out of the logarithm, making expressions easier to work with. Another useful identity is that for the natural logarithm of \( e \), our base: \( \ln(e) = 1 \). This identity simplifies computations since any expression involving \( \ln(e^x) \) simply becomes \( x \).
These properties allow us to break down and simplify logarithmic expressions into basic, manageable components.
Logarithmic identities are not just confined to base \( e \). However, natural logarithms are unique because they always use base \( e \), which is approximately 2.718. These identities give us the tools needed to simplify and compute expressions more quickly and accurately.

Understanding Exponentiation

Exponentiation involves raising a number, known as the base, to a certain power, called the exponent. In our exercise, the expression \( \sqrt[4]{e} \) is the fourth root of the mathematical constant \( e \).
To express the fourth root as an exponent, we write \( e^{1/4} \). This is because taking the nth root of a number is equivalent to raising it to the power of \( 1/n \).

Here are some fundamental points about exponents:

• Exponentiation is the process of multiplying a number by itself a certain number of times.
• An exponent of 1/4 means the fourth root of the base number.
• Using exponent rules helps to solve expressions involving roots or powers.

Exponentiation is a key concept in mathematics that bridges the understanding between roots and powers.
Mastering exponent rules allows for efficient manipulation of expressions, especially when dealing with complex operations such as roots.

Simplification in Logarithmic Expressions

Simplification is the process of reducing an expression to its simplest form. With logarithms, it often involves using known identities and properties to rewrite expressions in a simpler, more compact form. In our exercise about \( \ln \sqrt[4]{e} \), simplification begins by identifying the fourth root as an exponent: \( e^{1/4} \).
From there, we leverage the logarithmic power rule: \( \ln(a^b) = b \ln(a) \). Applying this rule simplifies the expression to \( \frac{1}{4} \ln(e) \).
Knowing \( \ln(e) = 1 \), we can further simplify it to \( \frac{1}{4} \), which is the simplest form. Simplification involves:

• Identifying powers and roots as exponents
• Applying logarithmic identities to condense expressions
• Using known values, like \( \ln(e) \), to further reduce complexity

Simplification is advantageous as it untangles more complex expressions into straightforward, calculable results,
making calculations less daunting and more intuitive.