Problem 50 - Intermediate Algebra

Question

Evaluate each expression without using a calculator. $$ \ln \sqrt[4]{e^{3}} $$

Step-by-Step Solution

Answer

The expression evaluates to \( \frac{3}{4} \).

Step 1: Simplify the expression inside the logarithm

We start by simplifying the expression under the logarithm. The expression is \( \sqrt[4]{e^3} \). The fourth root of \( e^3 \) can be rewritten using exponents as \( (e^3)^{1/4} \).

Step 2: Apply the power rule for exponents

The power rule for exponents states that \( (a^m)^n = a^{m \times n} \). We apply this to \( (e^3)^{1/4} \):\[ (e^3)^{1/4} = e^{3 \times 1/4} = e^{3/4}. \]

Step 3: Apply the logarithmic identity

Using the identity \( \ln(a^b) = b \ln a \), we can take the natural logarithm of \( e^{3/4} \):\[ \ln(e^{3/4}) = \frac{3}{4} \ln e. \]

Step 4: Evaluate the natural logarithm

Since \( \ln e = 1 \), we substitute this value into the expression from Step 3:\[ \frac{3}{4} \ln e = \frac{3}{4} \times 1 = \frac{3}{4}. \]

Key Concepts

Logarithmic Identities

Logarithmic identities are essential tools that help in simplifying and evaluating logarithmic expressions. One of the most useful identities is the power rule, which states that \( \ln(a^b) = b \ln a \). This property allows us to bring down the exponent as a multiplier, making calculations much easier.
For example, in the expression \( \ln(e^{3/4}) \), the power rule helps us rewrite it as \( \frac{3}{4} \ln e \). Another important identity is that the natural logarithm of \( e \) is 1, i.e., \( \ln e = 1 \). This makes the evaluation straightforward in our exercise, ultimately leading to \( \ln(e^{3/4}) = \frac{3}{4} \ln e = \frac{3}{4} \).
Understanding these identities simplifies working with logarithms, especially when evaluating logarithmic expressions without a calculator. These are effective shortcuts that save time and effort across various mathematical problems.

Exponents

Exponents represent repeated multiplication and are a fundamental part of algebra. The expression \( (e^3)^{1/4} \) involves exponents and can be simplified using basic laws known as the power rule for exponents.
This rule states that \( (a^m)^n = a^{m \times n} \). By applying this to \( (e^3)^{1/4} \), we multiply the exponents: \( e^{3 \times 1/4} = e^{3/4} \). This simplification step is necessary in preparing the expression for further evaluation.
Exponents help in expressing repeated multiplication in a compact form, making complex calculations manageable. Knowing how to manipulate exponents using rules like the power rule is pivotal in simplifying expressions both in everyday situations and higher-level mathematics.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form without changing their value. It's a skill that makes complex and lengthy expressions easier to understand and solve.
In our exercise, simplifying starts with breaking down the expression inside the natural logarithm: \( \sqrt[4]{e^3} \). By translating the fourth root into an exponent form, \( (e^3)^{1/4} \), we can apply the power rule for exponents, simplifying it further to \( e^{3/4} \).
This simplified form is crucial when proceeding to take the natural logarithm. Each step of simplification systematically removes complications from the original problem.
Overall, expression simplification utilizes a combination of rules and properties (like those of exponents and logarithms) and transforms complex expressions into straightforward ones, which is invaluable in both solving mathematical problems and enhancing conceptual understanding.