Problem 40 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$\text { In } \sqrt[3] e$$

Step-by-Step Solution

Answer

\( \frac{1}{3} \)

Step 1: Understand the Problem

The problem requires simplification of the expression \( \ln \sqrt[3]{e} \). This involves both a natural logarithm (\( \ln \)) and a cube root.

Step 2: Rewrite the Cube Root as an Exponent

Use the property of radicals to rewrite the cube root in exponential form. The cube root of \( e \) can be expressed as \( e^{1/3} \). Thus, the expression becomes \( \ln(e^{1/3}) \).

Step 3: Apply the Logarithm Power Rule

Using the logarithm power rule, \( \ln(a^b) = b \cdot \ln(a) \), apply this to the expression \( \ln(e^{1/3}) \). This simplifies to \( \frac{1}{3} \cdot \ln(e) \).

Step 4: Use the Property of the Natural Logarithm

Recall that \( \ln(e) = 1 \) because the natural logarithm of \( e \) is one by definition. Substitute 1 into the expression: \( \frac{1}{3} \cdot 1 = \frac{1}{3} \).

Step 5: Simplification Complete

The simplified expression of \( \ln \sqrt[3]{e} \) is \( \frac{1}{3} \).

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a special logarithm with the base \( e \), where \( e \) is approximately equal to 2.71828. This constant is fundamental in mathematics and appears in various natural processes. The natural logarithm has a unique property: \( \ln(e) = 1 \). This means if you take the natural log of \( e \), it simplifies to 1.
In our example \( \ln \sqrt[3]{e} \), recognizing this property helps simplify the expression down substantially. Understanding that \( \ln(e) = 1 \) allows us to reduce more complicated expressions involving this operation easily by using such identities.

Logarithm Power Rule

The logarithm power rule is a very helpful property for simplifying expressions involving logarithms, especially when the argument of the logarithm is raised to a power. The rule states that \( \ln(a^b) = b \cdot \ln(a) \). This means that the exponent can be brought out to the front as a multiplier.
For example, in \( \ln(e^{1/3}) \), we leverage the logarithm power rule by moving the exponent \( \frac{1}{3} \) to the front, resulting in \( \frac{1}{3} \times \ln(e) \). As we have learned from the property of the natural logarithm, \( \ln(e) = 1 \). Therefore, the expression simplifies to \( \frac{1}{3} \times 1 = \frac{1}{3} \). This simplification greatly reduces the complexity of handling powers inside logarithms.

Radicals

Radicals involve root operations, such as square roots or cube roots. The radical sign is used to denote these operations. For example, the cube root of a number \( x \) is written as \( \sqrt[3]{x} \), which can be expressed in exponential form as \( x^{1/3} \).
When working with radicals in logarithms, converting them to their exponential form is often helpful. In our case with \( \ln \sqrt[3]{e} \), we rewrite the cube root as \( e^{1/3} \). This transformation makes it easier to apply logarithmic rules, such as the power rule, for simplification. Being comfortable with transitioning between radical and exponential forms is beneficial when tackling algebraic expressions involving roots.