Problem 62 - Intermediate Algebra

Question

Write each logarithm as the sum and/or difference of logarithms of a single quantity. Then simplify, if possible. See Example 6. $$ \log _{3} \frac{\sqrt[4]{x}}{y z} $$

Step-by-Step Solution

Answer

\( \log_3 \left( \frac{\sqrt[4]{x}}{yz} \right) = \frac{1}{4} \log_3(x) - \log_3(y) - \log_3(z). \)

Step 1: Apply the Quotient Rule of Logarithms

The quotient rule states that \( \log_b \left( \frac{M}{N} \right) = \log_b(M) - \log_b(N) \). In this expression, we have \( M = \sqrt[4]{x} \) and \( N = yz \). Apply the quotient rule to get: \[ \log_3 \left( \frac{\sqrt[4]{x}}{yz} \right) = \log_3 (\sqrt[4]{x}) - \log_3(yz). \]

Step 2: Apply the Power Rule to the First Term

The power rule of logarithms states that \( \log_b (M^n) = n \log_b(M) \). Here, \( \sqrt[4]{x} \) can be rewritten as \( x^{1/4} \). Thus, use the power rule: \[ \log_3 (\sqrt[4]{x}) = \log_3 (x^{1/4}) = \frac{1}{4} \log_3(x). \]

Step 3: Apply the Product Rule of Logarithms to the Second Term

The product rule for logarithms states that \( \log_b(MN) = \log_b(M) + \log_b(N) \). In \( \log_3(yz) \), set \( M = y \) and \( N = z \). Applying the product rule gives: \[ \log_3(yz) = \log_3(y) + \log_3(z). \]

Step 4: Combine All Calculated Logarithms

Substitute the results from the previous steps into the expression from Step 1. We have: \[ \log_3 \left( \frac{\sqrt[4]{x}}{yz} \right) = \frac{1}{4} \log_3(x) - (\log_3(y) + \log_3(z)). \] This simplifies to: \[ \log_3 \left( \frac{\sqrt[4]{x}}{yz} \right) = \frac{1}{4} \log_3(x) - \log_3(y) - \log_3(z). \]

Key Concepts

Quotient Rule of Logarithms

The quotient rule of logarithms is a handy tool whenever you need to separate the logarithm of a division into simpler parts. This rule states that \( \log_b \left( \frac{M}{N} \right) = \log_b(M) - \log_b(N) \). In simple terms, it means that the logarithm of a fraction is the same as the difference between the logarithms of the numerator and the denominator.
In our original expression \( \log_3 \left( \frac{\sqrt[4]{x}}{yz} \right) \), think of \( \sqrt[4]{x} \) as "M" and \( yz \) as "N". Apply the quotient rule here to break the expression into two separate logarithms: \( \log_3(\sqrt[4]{x}) - \log_3(yz) \). This decomposition is the first step to simplifying the big picture.
This operation makes complex expressions easier to manage and is crucial when simplifying or expanding logarithmic expressions. Using this rule, you can transform complex divisions into subtraction problems, which are often simpler to work with.

Power Rule of Logarithms

The power rule of logarithms is another essential tool in your logarithmic arsenal. It states that \( \log_b (M^n) = n \log_b(M) \). This rule allows you to move a power out in front of the logarithm as a coefficient, thus simplifying the expression.
For our task, we need to apply the power rule to \( \log_3(\sqrt[4]{x}) \). Realize that \( \sqrt[4]{x} \) is simply \( x^{1/4} \) in disguise. So, according to the power rule, \( \log_3(x^{1/4}) \) becomes \( \frac{1}{4} \log_3(x) \).
Understanding and using the power rule helps you transform and simplify expressions that involve roots or exponents. This can be incredibly useful when dealing with equations where roots or powers might complicate things unnecessarily.

Product Rule of Logarithms

Last but not least, the product rule for logarithms helps tackle expressions where multiplication is involved. It states that \( \log_b(MN) = \log_b(M) + \log_b(N) \). By using this rule, you can break down products within a logarithm into a sum of individual logarithms.
Applied to our expression, \( \log_3(yz) \), we consider "y" as \( M \) and "z" as \( N \). Using the product rule, we expand this into \( \log_3(y) + \log_3(z) \).
This breaking down into sums simplifies otherwise complex multiplication situations within logarithms. It is particularly helpful in problems that require simplifying multiplied terms into separate components.