Problem 20 - Algebra and Trigonometry

Question

Use the Laws of Logarithms to expand the expression. $$ \log _{6} \sqrt[4]{17} $$

Step-by-Step Solution

Answer

The expanded expression is \( \frac{1}{4} \log_6(17) \).

Step 1: Express the Radical as an Exponent

Recall that the fourth root of a number can be expressed as a power, specifically to the power of \( \frac{1}{4} \). Therefore, you can rewrite \( \sqrt[4]{17} \) as \( 17^{\frac{1}{4}} \).

Step 2: Apply the Power Rule of Logarithms

The Power Rule states that \( \log_b(a^n) = n \cdot \log_b(a) \). By applying this rule, the expression becomes \( \frac{1}{4} \cdot \log_6(17) \).

Step 3: Rewrite the Expression

After applying the Power Rule, the expanded version of the logarithmic expression is \( \frac{1}{4} \log_6(17) \). The expression is now fully expanded.

Key Concepts

Expansion of Logarithmic Expressions

When faced with a logarithmic expression, such as \( \log _{6} \sqrt[4]{17} \), the goal of expansion is to express it in simpler parts that are easier to understand or work with mathematically. This can often involve breaking down complex expressions into simpler logs or constants.
To start, we need to recognize how to translate the original expression into a more manageable form using mathematical operations. Here, converting the root into an exponent by recognizing that \( \sqrt[4]{17} \) is the same as \( 17^{\frac{1}{4}} \) helps us proceed to use the laws of logarithms effectively.
This approach is particularly useful for computations that might otherwise seem intimidating. By expanding logarithmic expressions, you convert multiplicative relationships into exponential, making calculations more straightforward.

Power Rule of Logarithms

The Power Rule of Logarithms is a handy tool that transforms expressions with exponents into a simpler form. This rule is stated as: \( \log_b(a^n) = n \cdot \log_b(a) \). This means that when you have a logarithm with an exponent, you can "bring down" that exponent as a multiplier in front of the logarithm.
This rule is what allows us to shift from the expression \( \log_6(17^{\frac{1}{4}}) \), to the expanded form of \( \frac{1}{4} \cdot \log_6(17) \). The beauty of the power rule is its simplicity—rather than handling the complex expression inside the logarithm, you multiply the logarithm by the power instead.
Using this rule helps in simplifying equations, making it easier to solve for unknowns or perform further manipulations. It's a fundamental part of working with logarithmic functions efficiently.

Logarithmic Expressions

Logarithmic expressions are mathematical phrases that involve logs, which are operations that help us determine what power a base number is raised to in order to obtain another number. In simple terms, a logarithm answers the question: "To what power must the base be raised, to produce this number?"
In our example, \( \log_6(17) \) is the logarithm of the number 17 with a base of 6. Logarithms come with their own set of rules or laws, which include the famous Power, Product, and Quotient rules, each helping in simplifying and expanding expressions.
Grasping the concept of logarithmic expressions is crucial because they appear frequently in contexts ranging from algebra to calculus. They can simplify complex calculations, provide clearer insight into the behavior of functions, and are integral to understanding exponential growth and decay processes.