Problem 36 - Algebra and Trigonometry

Question

Use the Laws of Logarithms to expand the expression. $$ \log \sqrt{x \sqrt{y \sqrt{z}}} $$

Step-by-Step Solution

Answer

\( \frac{1}{2}\log x + \frac{1}{4}\log y + \frac{1}{8}\log z \).

Step 1: Converting Square Roots to Fractional Exponents

The given expression is \( \log \sqrt{x \sqrt{y \sqrt{z}}} \). The first step is to express the square roots as fractional exponents. We can rewrite \( \sqrt{x} \) as \( x^{1/2} \), \( \sqrt{y} \) as \( y^{1/2} \), and \( \sqrt{z} \) as \( z^{1/2} \). Therefore, the expression \( x \sqrt{y \sqrt{z}} \) becomes \( x(yz^{1/2})^{1/2} \), which simplifies to \( x^{1/2}(y^{1/2})(z^{1/4}) \) using laws of exponents.

Step 2: Applying the Power Rule

The power rule of logarithms states that \( \log(a^b) = b \log(a) \). Applying this to the expression \( \log(x^{1/2} y^{1/4} z^{1/8}) \), we get: \[\log(x^{1/2} y^{1/4} z^{1/8}) = \frac{1}{2}\log x + \frac{1}{4}\log y + \frac{1}{8}\log z.\]

Step 3: Simplifying the Expression

The expression \( \frac{1}{2}\log x + \frac{1}{4}\log y + \frac{1}{8}\log z \) is expanded using the logarithmic rules and cannot be simplified further, as it is the fully expanded form.

Key Concepts

Laws of Logarithms

Logarithms are incredibly useful tools in mathematics, and understanding the laws governing them is essential. The laws of logarithms make it easier to manipulate logarithmic expressions, especially when we need to simplify or expand them. Three key laws of logarithms are commonly used: the product rule, the quotient rule, and the power rule.

1. **Product Rule**: This rule states that the logarithm of a product is the sum of the logarithms. Mathematically expressed as:

• \(\log(ab) = \log(a) + \log(b).\)

2. **Quotient Rule**: This says that the logarithm of a quotient is the difference of the logarithms:

• \(\log\left(\frac{a}{b}\right) = \log(a) - \log(b).\)

3. **Power Rule**: This useful rule indicates that the logarithm of a power is the exponent multiplied by the logarithm of the base, as in:

• \(\log(a^b) = b \cdot \log(a).\)

These rules simplify many problems that involve logarithms, and knowing how to apply them correctly is a valuable skill in solving logarithmic equations. In our exercise, the power rule played a pivotal role.

Expanding Expressions

Expanding logarithmic expressions involves transforming a complex logarithmic expression into a simpler, more manageable form. This usually requires applying the laws of logarithms to break down the expression into simpler parts.

The goal is to separate the logarithm of a composite expression like a product, quotient, or power into a sum or difference of simpler logs. Each component is dealt with separately to make calculation or further manipulation straightforward. In our more complex expression \(\log \sqrt{x \sqrt{y \sqrt{z}}} \)

• First, we transform the nested square roots into individual fractional powers.
• Next, we use the power rule to expand the expression further.

After applying these steps, we can express the single compound logarithm as a sum of simple logarithms, providing clarity and ease of computation.

Power Rule of Logarithms

The power rule of logarithms is a powerful tool when dealing with logarithms that have exponents. This rule states that when you have a logarithm of an expression raised to a power, you can move the exponent in front of the logarithm as a multiplier.

Mathematically, it looks like this:

• \(\log(a^b) = b \times \log(a).\)

In the context of our exercise, this rule simplifies our initial complex expression. We transformed the square roots of each variable into fractional exponents, ultimately allowing us to apply the power rule. For example, in the term \(x^{1/2}\), this term becomes \(\frac{1}{2} \log x\).

• This simplification is critical in expanding the expression finely.
• Without the power rule, handling such expressions would be far more challenging.

Understanding this rule enhances our ability to decompose logarithmic expressions into their simplest forms.

Fractional Exponents

Fractional exponents are an alternative way to represent roots using exponents. Converting radicals or roots into fractional exponents is often necessary, especially when dealing with logarithmic expressions.

For instance, the square root of a variable \(x\) can be written as \(x^{1/2}\). Similarly, the cube root of \(y\) is expressed as \(y^{1/3}\). This method simplifies the handling of roots in complex mathematical expressions, especially when used alongside the laws of logarithms.

• In our exercise, we converted nested square roots into fractional exponents before expansion.
• Specifically, we expressed \(\sqrt{x\sqrt{y\sqrt{z}}}\) as \(x^{1/2}y^{1/4}z^{1/8}\).

This conversion is vital because logarithmic laws, like the power rule, work seamlessly with fractional exponents, allowing us to expand logarithmic expressions accurately. Mastering fractional exponents aids significantly in simplifying and manipulating mathematical expressions efficiently.