Problem 2 - Algebra and Trigonometry with Analytic Geometry

Question

Express in terms of logarithms of \(x, y, z\), or \(w\). (a) \(\log _{3}(x y z)\) (b) \(\log _{3}(x z / y)\) (c) \(\log _{3} \sqrt[5]{y}\)

Step-by-Step Solution

Answer

(a) \( \log_{3}(x) + \log_{3}(y) + \log_{3}(z) \), (b) \( \log_{3}(x) + \log_{3}(z) - \log_{3}(y) \), (c) \( \frac{1}{5} \cdot \log_{3}(y) \)."

Step 1: Applying the Product Rule for Logarithms

The product rule for logarithms states that \( \log_b(AB) = \log_b(A) + \log_b(B) \). For the expression \( \log_{3}(xyz) \), apply the product rule iteratively: \[ \log_{3}(xyz) = \log_{3}(x) + \log_{3}(y) + \log_{3}(z) \]

Step 2: Applying the Quotient Rule for Logarithms

The quotient rule for logarithms states that \( \log_b(\frac{A}{B}) = \log_b(A) - \log_b(B) \). For the expression \( \log_{3}(\frac{xz}{y}) \), apply this rule: \[ \log_{3}(\frac{xz}{y}) = \log_{3}(xz) - \log_{3}(y) \]Next, apply the product rule to \( \log_{3}(xz) \): \[ \log_{3}(xz) = \log_{3}(x) + \log_{3}(z) \]So, the expression becomes:\[ \log_{3}(\frac{xz}{y}) = \log_{3}(x) + \log_{3}(z) - \log_{3}(y) \]

Step 3: Applying the Power Rule for Logarithms

The power rule for logarithms states that \( \log_b(A^n) = n \cdot \log_b(A) \). For the expression \( \log_{3}(\sqrt[5]{y}) \), note that \( \sqrt[5]{y} = y^{1/5} \). Apply the power rule:\[ \log_{3}(y^{1/5}) = \frac{1}{5} \cdot \log_{3}(y) \]

Key Concepts

Product Rule for Logarithms

The product rule for logarithms is a very handy tool when dealing with the logarithm of a product of numbers. It allows us to simplify this into a sum of individual logarithms. Essentially, this rule states that the logarithm of a product is equal to the sum of the logarithms of its factors. The formula is as follows:\[ \log_b(AB) = \log_b(A) + \log_b(B) \]This can be extended to more than two factors. For example, if you have three variables, say \(x, y,\) and \(z\), and you're trying to find \(\log_{3}(xyz)\), you would apply the product rule to find:\[ \log_{3}(xyz) = \log_{3}(x) + \log_{3}(y) + \log_{3}(z) \]This transformation makes larger expressions easier to handle as it breaks down the multiplicative components into simpler, additive parts.

• Useful for simplifying the multiplication of variables or expressions into separate logs.
• Makes calculations more straightforward when dealing with logarithms in algebra.

Quotient Rule for Logarithms

When you come across a division inside a logarithm, the quotient rule comes to the rescue. This rule states that the logarithm of a fraction is equivalent to the logarithm of the numerator minus the logarithm of the denominator, expressed in the following formula:\[ \log_b\left(\frac{A}{B}\right) = \log_b(A) - \log_b(B) \]Consider the expression \(\log_{3}\left(\frac{xz}{y}\right)\). By using the quotient rule, we first break it into:\[ \log_{3}(xz) - \log_{3}(y) \]Next, apply the product rule to further split \(\log_{3}(xz)\) into:\[ \log_{3}(x) + \log_{3}(z) \]Combining these steps, the complete expression becomes:\[ \log_{3}(x) + \log_{3}(z) - \log_{3}(y) \]

• Ideal for handling divisions within logarithms to simplify expressions before calculations.
• Transforms complex divisions into simpler problems by separating into subtraction of logs.

Power Rule for Logarithms

The power rule for logarithms simplifies expressions where a number is raised to an exponent within a logarithmic expression. This rule tells us that the exponent can be moved in front of the logarithm, multiplying it by the logarithm. It is expressed mathematically as:\[ \log_b(A^n) = n \cdot \log_b(A) \]Taking the expression \(\log_{3}(\sqrt[5]{y})\) as an example, recognize that a fifth root can be rewritten using exponents: \(y^{1/5}\). Hence, applying the power rule yields:\[ \log_{3}(y^{1/5}) = \frac{1}{5} \cdot \log_{3}(y) \]This allows for simplification by reducing the complexity of the original expression, making it more manageable for further mathematical operations.

• Enables easy manipulation of exponential terms within logarithms.
• Vital for solving equations involving powers within logarithmic contexts efficiently.