Problem 26 - Algebra and Trigonometry

Question

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \log _{6} \sqrt[4]{17} $$

Step-by-Step Solution

Answer

\( \frac{1}{4} \log_{6}(17) \)

Step 1: Identify the Logarithm Expression

The expression we need to expand is \( \log_{6} \sqrt[4]{17} \). Our goal is to use the laws of logarithms to express this in a more expanded form.

Step 2: Convert the Radical to Exponential Form

The fourth root, \( \sqrt[4]{17} \), can be rewritten using exponent notation as \( 17^{1/4} \). So, \( \log_{6} \sqrt[4]{17} \) becomes \( \log_{6} (17^{1/4}) \).

Step 3: Apply the Power Rule of Logarithms

The power rule states that \( \log_{b}(x^{n}) = n \cdot \log_{b} (x) \). Applying this to \( \log_{6} (17^{1/4}) \), we get \( \frac{1}{4} \cdot \log_{6}(17) \).

Step 4: Finalize the Expanded Expression

The expanded form of the logarithmic expression is \( \frac{1}{4} \log_{6}(17) \). This is the expression in its fully expanded form based on the laws of logarithms.

Key Concepts

Logarithmic Expressions

Logarithms are a fascinating mathematical concept that serve as the inverse of exponentiation. When you see a logarithmic expression, it generally looks something like this: \( \log_{b}(x) \), where \( b \) is the base and \( x \) is the value you’re taking the logarithm of.
Logarithms help us deal with large numbers and exponential relationships in a manageable way.
In practice, logarithmic expressions allow us to transform multiplication into addition and exponentiation into multiplication. This transformation makes calculations easier, especially when handling complex equations. When you're asked to expand a logarithmic expression, your goal is to convert it into a simpler form using the various laws of logarithms.
These laws include:

• The Product Rule: \( \log_{b}(mn) = \log_{b}(m) + \log_{b}(n) \)
• The Quotient Rule: \( \log_{b}(\frac{m}{n}) = \log_{b}(m) - \log_{b}(n) \)
• The Power Rule: \( \log_{b}(m^{n}) = n \cdot \log_{b}(m) \)

These rules are essential tools for expanding and simplifying logarithmic expressions.

Power Rule of Logarithms

The power rule of logarithms is one of the most important tools for simplifying logarithmic expressions. This rule states that if you have a logarithm of a power, such as \( \log_{b}(x^{n}) \), you can bring the exponent \( n \) down in front as a multiplier, transforming it into \( n \cdot \log_{b}(x) \).
This simplification makes calculations easier, especially when dealing with complex bases and large numbers.
Why is this helpful? Here are some reasons:

• It simplifies the handling of variables raised to powers.
• It breaks down complex calculations into simpler, linear parts.

In the provided exercise, we applied this rule to the expression \( \log_{6} (17^{1/4}) \). By using the power rule, it simplified to \( \frac{1}{4} \cdot \log_{6}(17) \), making the expression easier to understand and work with.

Radical to Exponential Conversion

Converting radicals to exponential form is a crucial step when working with logarithms. Radicals, like square roots or fourth roots, can be represented using exponents, which makes it possible to apply logarithmic rules.
This conversion is essential because many rules of logarithms rely on exponent format rather than on radical notation.
To convert a radical into an exponential form:

• A square root, such as \( \sqrt{x} \), becomes \( x^{1/2} \).
• The fourth root, such as \( \sqrt[4]{x} \), becomes \( x^{1/4} \).

In our exercise, the fourth root \( \sqrt[4]{17} \) was converted to exponent notation as \( 17^{1/4} \). This conversion facilitated the use of the power rule of logarithms, allowing for the simplification of the entire expression. With this method, you're always able to move from radicals to a format ready for further manipulation using logarithmic rules.