Problem 54 - Elementary and Intermediate Algebra

Question

Write logarithm without an exponent or a radical symbol. Then simplify, if possible. \(\log \sqrt[3]{7}\)

Step-by-Step Solution

Answer

\( \log \sqrt[3]{7} = \frac{1}{3} \cdot \log 7 \)

Step 1: Understand the Radical Notation

The original logarithmic expression contains a cube root: \( \log \sqrt[3]{7} \). A cube root can also be expressed as an exponent of \( \frac{1}{3} \). So we can rewrite: \( \sqrt[3]{7} \) as \( 7^{\frac{1}{3}} \). Substitute it into the logarithm: \( \log 7^{\frac{1}{3}} \).

Step 2: Apply the Power Rule of Logarithms

The power rule for logarithms states \( \log b^a = a \cdot \log b \). Applying this rule to our expression, we get: \( \log 7^{\frac{1}{3}} = \frac{1}{3} \cdot \log 7 \).

Step 3: Simplify the Expression

The expression \( \frac{1}{3} \cdot \log 7 \) cannot be simplified further without additional numeric information on \( \log 7 \). This is the simplest form of the logarithmic expression.

Key Concepts

Radical Notation

Radical notation is often encountered in expressions that involve roots, such as square roots, cube roots, and beyond. It is represented by the radical symbol \( \sqrt{} \), and each type of root has its specific notation. For example, a square root is denoted by \( \sqrt{a} \), whereas a cube root is written as \( \sqrt[3]{a} \). These symbols denote the operations needed to find a particular root of a number, where cube roots find a number that when multiplied by itself three times equals the original number.

When dealing with logarithms and radicals together, you might need to convert radical notation into an exponent form. Converting \( \sqrt[3]{7} \) into an exponent gives us \( 7^{\frac{1}{3}} \). This transformation is helpful in simplifications, especially within logarithmic expressions where exponent rules can be more easily applied.

Exponent Rules

Exponent rules are essential in simplifying expressions involving powers. They help manipulate and reduce expressions to simpler forms. Important exponent rules include the power of a power rule, the product of powers rule, and the power of a quotient rule.

In the context of logarithms, the power rule is particularly useful: \( \log b^a = a \cdot \log b \). This rule allows us to bring the exponent of a number out in front of the logarithm, effectively linearizing it into a multiplication, simplifying expressions considerably. For instance, with our task of rewriting \( \log \sqrt[3]{7} \) as \( \log 7^{\frac{1}{3}} \), the power rule allows us to extract \( \frac{1}{3} \) from the exponent, resulting in the expression \( \frac{1}{3} \cdot \log 7 \).

Cube Roots

Cube roots are a specific type of root that are vital in various mathematical operations, including simplifying equations and solving problems in algebra. The cube root of a number \( a \) is the number which, when multiplied by itself twice more (i.e., cubed), results in \( a \). In mathematical notation, this is expressed as \( \sqrt[3]{a} \).

Understanding cube roots is particularly important when you encounter them in expressions like \( \log \sqrt[3]{7} \). Here, identifying that \( \sqrt[3]{7} \) can be written as \( 7^{\frac{1}{3}} \) is the first step in simplifying an expression and applying further mathematical rules, like those involving exponents or logarithms, to achieve a simplified form such as \( \frac{1}{3} \log 7 \). This insight simplifies dealing with complex expressions involving cube roots.