Problem 68 - Elementary and Intermediate Algebra

Question

Write logarithm as the sum and/or difference of logarithms of a single quantity. Then simplify, if possible. \(\log _{b} \sqrt[4]{\frac{x^{3} y^{2}}{z^{4}}}\)

Step-by-Step Solution

Answer

\( \frac{3}{4}\log_{b}(x) + \frac{1}{2}\log_{b}(y) - \log_{b}(z) \)

Step 1: Convert the radical to an exponent

The fourth root of a quantity, \( \sqrt[4]{A} \), can be expressed as an exponent: \( A^{1/4} \). Apply this to the expression \( \sqrt[4]{\frac{x^{3} y^{2}}{z^{4}}} \), resulting in \( \left(\frac{x^{3}y^{2}}{z^{4}}\right)^{1/4} \).

Step 2: Apply logarithm power rule

The power rule for logarithms states that \( \log_{b}(A^{n}) = n \cdot \log_{b}(A) \). Apply this to the expression, resulting in: \( \frac{1}{4} \cdot \log_{b}\left(\frac{x^{3}y^{2}}{z^{4}}\right) \).

Step 3: Apply logarithm quotient rule

The quotient rule for logarithms states that \( \log_{b} \left( \frac{A}{B} \right) = \log_{b}(A) - \log_{b}(B) \). Apply this to \( \log_{b}\left(\frac{x^{3}y^{2}}{z^{4}}\right) \), resulting in: \( \log_{b}(x^{3}y^{2}) - \log_{b}(z^{4}) \).

Step 4: Apply logarithm product rule

The product rule for logarithms states that \( \log_{b}(AB) = \log_{b}(A) + \log_{b}(B) \). Apply this to \( \log_{b}(x^{3}y^{2}) \), resulting in: \( \log_{b}(x^{3}) + \log_{b}(y^{2}) \).

Step 5: Simplify each logarithm using the power rule

For each term, apply the power rule: \( \log_{b}(x^{3}) = 3\log_{b}(x) \) and \( \log_{b}(y^{2}) = 2\log_{b}(y) \). For the second part: \( \log_{b}(z^{4}) = 4\log_{b}(z) \).

Step 6: Combine and simplify the expression

Substitute the logs back in, resulting in: \[ \frac{1}{4} \left( 3\log_{b}(x) + 2\log_{b}(y) - 4\log_{b}(z) \right) \]. Distribute \( \frac{1}{4} \) to get: \( \frac{3}{4}\log_{b}(x) + \frac{1}{2}\log_{b}(y) - \log_{b}(z) \).

Key Concepts

Properties of Logarithms

Logarithms have unique properties that can help us break down or simplify expressions involving exponents and roots. Understanding these properties is essential to manipulate logarithmic equations effectively.

The three main properties are:

• Product Property: This states that the logarithm of a product is the sum of the logarithms of its factors. Mathematically, it's written as \( \log_{b}(AB) = \log_{b}(A) + \log_{b}(B) \). This property allows us to split products into simpler log terms.
• Quotient Property: Similar to the product property, this declares that the logarithm of a quotient is the difference between the logarithms of the numerator and the denominator. It's expressed as \( \log_{b} \left( \frac{A}{B} \right) = \log_{b}(A) - \log_{b}(B) \), enabling us to handle division inside logarithms by using subtraction.
• Power Property: This property indicates that the logarithm of a power can be rewritten by bringing the exponent out front as a multiplier. Formally, it's \( \log_{b} (A^{n}) = n \cdot \log_{b}(A) \). This property helps simplify expressions where the argument of the log is raised to an exponent.

These properties are crucial for simplifying complex logarithmic expressions, like the initial problem, by reducing them into more manageable parts.

Logarithm Rules

Logarithm rules are vital shortcuts that allow us to efficiently solve and manipulate logarithmic expressions. Here are some basic rules that mirror their exponential counterparts.

• The Identity Rule: This rule states \( \log_{b}(b) = 1 \). It arises because raising the base \( b \) to the power of 1 gives \( b \).
• The Zero Rule: It tells us that \( \log_{b}(1) = 0 \) since any base \( b \) to the power of 0 is equal to 1.
• Base Change Formula: Sometimes, it's useful to convert a logarithm to a different base, accomplished with the formula \( \log_{b}(A) = \frac{\log_{c}(A)}{\log_{c}(b)} \), for any positive base \( c \).

By combining these rules with the logarithmic properties, you can maneuver through various problems more easily.

In the original exercise, specifically, these rules justified the steps where expressions were converted from roots and fractions into forms using logarithmic subtraction and addition.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions is about breaking down complex problems into smaller, more manageable tasks by using logarithmic properties and rules.

In the given exercise, the solution to \( \log _{b} \sqrt[4]{\frac{x^{3} y^{2}}{z^{4}}} \) involved several key simplifications:

• First, the root was turned into a fraction exponent, using the equivalence \( \sqrt[4]{A} = A^{1/4} \). This step is crucial since logarithmic functions handle exponents neatly using the power property.
• Next, the problem was a division under the square root, which was made simpler by using the quotient property, splitting the logarithm of a fraction into the difference of two logs.
• Finally, each remaining product and power was tackled by utilizing the product and power properties, respectively. This helped refine each part of the expression into its simplest logarithmic form.

Such simplifications hinge on understanding and applying these core principles, making intricate expressions much easier to interpret and solve.