Problem 50 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The value is \(\frac{3}{4}\).

Step 1: Simplify the Expression

Start by expressing the fourth root as a power. The expression \( \sqrt[4]{e^3} \) is equivalent to \( (e^3)^{1/4} \).

Step 2: Apply the Power of a Power Property

Use the power of a power property, \((a^m)^n = a^{m \cdot n}\). Thus, \((e^3)^{1/4} = e^{3 \cdot (1/4)} = e^{3/4}\).

Step 3: Calculate the Natural Logarithm

The natural logarithm of \(e^x\) is simply \(x\). Therefore, \(\ln(e^{3/4}) = 3/4\).

Key Concepts

Powers and Roots

Understanding powers and roots is essential to simplify mathematical expressions. When dealing with roots, you can express them as fractional exponents. This approach helps make further calculations easier, especially when logarithms or exponents are involved. For example, the fourth root of any number or expression, denoted as \( \sqrt[4]{a} \), can be rewritten as \( a^{1/4} \).

To illustrate, if you have \( \sqrt[4]{e^3} \), you express the radical expression as a power by raising \(e^3\) to the \(1/4\) power. This conversion simplifies the expression into a form that aligns with broader mathematical rules, making subsequent steps like applying exponents or logarithmic properties much more straightforward. Recognizing and converting roots into fractional powers is a handy technique that often simplifies complex calculations.

Exponents

Exponents are a way to represent repeated multiplication of a number by itself. For example, \(3^4\) means \(3\) multiplied by itself four times: \(3 \times 3 \times 3 \times 3\). When expressions have exponents with another exponent, like a power raised to another power, you can use the power of a power property.

This property states that \((a^m)^n = a^{m \cdot n}\). With this property, the expression \((e^3)^{1/4}\) can be simplified to \(e^{3 \cdot 1/4} = e^{3/4}\).

• The base remains unchanged, \(e\) in this case.
• The exponents are multiplied together.
• This property is vital in simplifying expressions where a single base is involved in compounded exponents.

Understanding the power of a power property ensures that you can efficiently work through expressions with multiple layers of exponents.

Properties of Logarithms

Logarithms are powerful tools in mathematics. They serve as the inverse operation of exponentiation. The natural logarithm, denoted as \(\ln\), specifically relates to the base \(e\). One important property of logarithms, particularly the natural logarithm, is that it reverses exponentiation.

The property \(\ln(e^x) = x\) is fundamental. This property means if you take the natural logarithm of \(e\) raised to any power, the result is simply the exponent, \(x\). In practice, this simplifies evaluating expressions like \(\ln(e^{3/4})\), which directly results in \(3/4\).

• This property significantly simplifies complex expressions to manageable numbers.
• It shows the reciprocal nature of logarithms and powers, working together to resolve expressions effortlessly.
• Utilizing these basic properties is key to solving problems involving logarithms efficiently.