Problem 34 - Algebra and Trigonometry

Question

Use the Laws of Logarithms to expand the expression. $$ \log \left(\frac{x}{\sqrt[3]{1-x}}\right) $$

Step-by-Step Solution

Answer

\( \log(x) - \frac{1}{3}\log(1-x) \)

Step 1: Use the Quotient Rule

The first law of logarithms we can apply is the quotient rule, which states that \( \log\left(\frac{a}{b}\right) = \log(a) - \log(b) \). In this case, the expression inside the logarithm is \( \frac{x}{\sqrt[3]{1-x}} \). Thus, we can expand it to: \[ \log(x) - \log(\sqrt[3]{1-x}) \]

Step 2: Simplify the Logarithm of the Denominator Using the Root Rule

Next, apply the law of logarithms for roots, which states that \( \log(a^n) = n\cdot\log(a) \). In our expression \( \sqrt[3]{1-x} = (1-x)^{1/3} \). Therefore, apply the root rule: \[ \log((1-x)^{1/3}) = \frac{1}{3} \cdot \log(1-x) \]

Step 3: Combine the Results

Combine the results from Step 1 and Step 2 to write the final expanded form of the given logarithmic expression. Using the intermediate results from the previous steps, the expression becomes: \[ \log(x) - \frac{1}{3}\log(1-x) \]

Key Concepts

Laws of Logarithms

Logarithms are incredibly useful in simplifying mathematical expressions, especially those involving exponentiation and multiplication. The Laws of Logarithms provide us with rules to manipulate logarithms conveniently. Here are a few important laws:

• Product Rule: This rule states that the logarithm of a product is the sum of the logarithms of its factors. Mathematically, it is expressed as \( \log(ab) = \log(a) + \log(b) \).
• Quotient Rule: This rule tells us how to handle division within a logarithm by converting it to subtraction. For any two positive numbers \( a \) and \( b \), \( \log\left(\frac{a}{b}\right) = \log(a) - \log(b) \).
• Power Rule: This simplifies the logarithm of an exponential expression. It says \( \log(a^n) = n \cdot \log(a) \).

Knowing these rules can help you break down complex logarithmic expressions into simpler components, making them easier to calculate or expand. By leveraging these laws, mathematical expressions, like the one in our exercise, can be written in a more understandable form.

Quotient Rule

The Quotient Rule is a key law of logarithms used to simplify expressions involving division inside a logarithm. This rule is incredibly helpful because it allows us to separate the numerator and the denominator into distinct logarithmic parts. For a fraction \( \frac{a}{b} \), the rule is written as:\[ \log\left(\frac{a}{b}\right) = \log(a) - \log(b) \].In our given exercise, we used the Quotient Rule to split the expression:\[ \log\left(\frac{x}{\sqrt[3]{1-x}}\right) = \log(x) - \log(\sqrt[3]{1-x}) \].This separation helps to further simplify and address each component independently. The advantage here is that it turns a complex division problem into a simpler subtraction problem involving two separate logarithmic terms.

Root Rule

When dealing with roots in logarithmic expressions, the Root Rule is extremely valuable. It simplifies expressions where the argument of the logarithm includes a root. This rule tells us that the root of a number can be expressed as an exponent. In mathematical terms:\( \sqrt[n]{a} = a^{1/n} \), and therefore \( \log(\sqrt[n]{a}) = \frac{1}{n} \cdot \log(a) \).In our exercise, we applied the Root Rule to the denominator \( \sqrt[3]{1-x} \), rewriting it as \( (1-x)^{1/3} \). This gave us:\[ \log((1-x)^{1/3}) = \frac{1}{3} \cdot \log(1-x) \].Using the Root Rule allows the expression to be expressed in terms of a simple logarithmic function, making it more straightforward to handle algebraically and ensuring that the expanded expression is clear.

Logarithmic Expansion

Logarithmic expansion involves using the laws of logarithms to break down a complex logarithmic expression into simpler parts. The goal is to convert a single logarithmic expression into a more manageable form using identifiable and simple individual components.In the given exercise, we sought to expand \( \log \left(\frac{x}{\sqrt[3]{1-x}}\right) \). Applying the Quotient Rule and then the Root Rule, the expression was thoroughly expanded into the final form:\[ \log(x) - \frac{1}{3}\log(1-x) \].This expanded form shows each constituent part of the initial logarithmic expression separately. Logarithmic expansion not only makes computations more approachable but also enhances comprehension of the underlying mathematical structure by exposing each part's contribution to the whole expression.