Problem 32 - Precalculus with Limits

Question

Using Properties of Logarithms In Exercises \(21-36,\) find the exact value of the logarithmic expression without using a calculator. (If this is not possible, then state the reason.) $$\ln \sqrt[4]{e^{3}}$$

Step-by-Step Solution

Answer

The exact value of the logarithmic expression \( \ln \sqrt[4]{e^{3}} \) is \( \frac{3}{4} \)

Step 1: Use properties of radicals

Let's rewrite the fourth root as a exponent expression using the rule \( \sqrt[n]{x^m} = x^{m/n} \). The given expression \( \ln \sqrt[4]{e^{3}} \) then becomes \( \ln {e^{3/4}} \)

Step 2: Apply logarithm properties

To simplify this further, we will use the property of logarithms that allows us to convert the exponent of the logarithm argument into a coefficient: \( \ln a^n = n \ln a \). Applying this rule, the expression \( \ln {e^{3/4}} \) simplifies to \( \frac{3}{4} \ln e \)

Step 3: Simplify the expression

We know that \( \ln e \) equals 1, since the natural logarithm \( \ln \) is logarithm base \( e \). Therefore, the expression simplifies to \( \frac{3}{4} \times 1 = \frac{3}{4} \)

Key Concepts

Logarithmic Expressions

Logarithms are a way to represent powers or exponents in mathematics. They are particularly useful for simplifying expressions and solving equations that involve exponential terms. The notation \( \ln \) is used to denote the natural logarithm, which has a base of \( e \), where \( e \approx 2.718 \). This is a vital constant in mathematics, often appearing in scenarios involving growth and decay.

One key property of logarithms is that logarithmic expressions can be manipulated using certain rules to make them easier to evaluate. The property we used in the original solution is that for any expression \( a \), \( \ln a^n = n \ln a \). This property allows us to "bring down" exponents as coefficients, making the calculation simpler. This makes working with logarithms particularly powerful when dealing with complex expressions. Understanding this property of logarithm will enhance your ability to approach various mathematical problems involving logs.

Simplifying Radical Expressions

Radical expressions often involve roots, like square roots or, as in our exercise, fourth roots. Simplifying these expressions can help in solving equations or in making more complex expressions manageable. Always remember a crucial property: \( \sqrt[n]{x^m} = x^{m/n} \). This allows you to transform a root into an exponent, which is often easier to handle in mathematical equations.

When simplifying radicals, it's also important to know how to recognize patterns and transform them into forms that are easier to work with. The transformation from a radical form \( \sqrt[4]{e^3} \) into an exponential form \( e^{3/4} \) is a great example of applying these simplification techniques. Mastering these properties aids in reducing expressions to simpler forms, particularly in logarithmic contexts where such transformations frequently arise.

Exponent Rules

Exponent rules are a cornerstone of algebra and higher mathematics, providing tools for simplifying expressions with powers. Some important rules include:

• Product of powers rule: \( a^m \cdot a^n = a^{m+n} \)
• Power of a power rule: \( (a^m)^n = a^{m \cdot n} \)
• Power of a product rule: \( (ab)^n = a^n \, b^n \)

In the specific exercise, we leveraged these rules when converting a radical form to an exponent by recognizing that \( e^3 \) raised to the fourth root is \( e^{3/4} \). Understanding how to manipulate and simplify expressions using these rules ultimately made solving the problem straightforward.

Knowing these rules helps efficiently deal with powers across different mathematical expressions, allowing for simplication and facilitating further operations like integration, differentiation, or solving equations.