Problem 21 - Algebra A Combined Function

Question

Find the exact value of each logarithm. $$ \ln \sqrt[4]{e} $$

Step-by-Step Solution

Answer

The exact value is \( \frac{1}{4} \).

Step 1: Rewrite the expression using radical form

The given expression is \( \ln \sqrt[4]{e} \). Rewrite \( \sqrt[4]{e} \) using an exponent: \( \sqrt[4]{e} = e^{1/4} \).

Step 2: Apply the Power Rule for Logarithms

The Power Rule for logarithms \( \ln(a^b) = b \cdot \ln(a) \) allows us to bring the exponent in front of the logarithm. Thus, \( \ln(e^{1/4}) = \frac{1}{4} \cdot \ln(e) \).

Step 3: Simplify using the value of \( \ln(e) \)

Since \( \ln(e) = 1 \), substitute this value into the expression: \( \frac{1}{4} \cdot 1 = \frac{1}{4} \).

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a logarithm with the base \( e \), where \( e \) is approximately equal to 2.71828. This number is known as Euler's number and is a fundamental constant in mathematics. The natural logarithm has unique properties that make it particularly useful in calculus and mathematical modeling.

• When you see \( \ln(e) \), it equals 1 because \( e^1 = e \).
• Natural logarithms are commonly used in growth and decay problems, as well as in solving equations involving exponentials.

Understanding \( \ln \) is crucial when you're dealing with natural processes, financial calculations, and any use involving exponential growth or decay. It simplifies expressions involving exponential functions, making them easier to solve or manipulate.

Radical Form

Radical form is a way of expressing roots, such as square roots or cube roots. In our example, we have the fourth root of \( e \), which is written as \( \sqrt[4]{e} \). By converting this radical expression into an exponent, it becomes \( e^{1/4} \).

• The index of the root (like the 4 in \( \sqrt[4]{ } \)) becomes the denominator of the fraction in the exponent.
• Radical form can be translated into exponential form to simplify operations like multiplication or to apply logarithm rules.

Switching between radical and exponent form is a great skill. It helps when you need to solve equations or simplify expressions involving roots.

Power Rule for Logarithms

The power rule for logarithms states that \( \ln(a^b) = b \cdot \ln(a) \). This rule is very handy because it allows us to move an exponent in front of the logarithm, simplifying the expression.

• In our example, \( \ln(e^{1/4}) \) allows us to bring \( 1/4 \) in front, leading to \( \frac{1}{4} \cdot \ln(e) \).
• This transformation helps in simplifying complex logarithmic expressions.

The power rule is applicable to any base, not just \( e \), making it versatile. It's a fundamental tool for solving logarithmic equations or simplifying expressions.

Exponents

Exponents are a way to represent repeated multiplication. The expression \( a^b \) means multiplying \( a \) by itself \( b \) times. Sometimes, exponents are fractions. Fractional exponents indicate roots, which can be converted to radical expressions and vice versa.

• An exponent of \( 1/4 \), for example, signifies the fourth root.
• Converting between radical form and exponent form helps in applying properties of exponents and logarithms easily.

Understanding exponents is key, as they naturally extend arithmetic into more challenging operations regarding growth, scaling, and roots. They are the backbone of exponential growth models and logarithmic functions.