Problem 40 - Algebra and Trigonometry

Question

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \log \left(\frac{x}{\sqrt[3]{1-x}}\right) $$

Step-by-Step Solution

Answer

The expanded expression is \( \log(x) - \frac{1}{3} \log(1-x) \).

Step 1: Apply the Quotient Rule for Logarithms

The quotient rule for logarithms states that \( \log \left( \frac{a}{b} \right) = \log(a) - \log(b) \). Apply this rule to the given expression: \( \log \left( \frac{x}{\sqrt[3]{1-x}} \right) = \log(x) - \log\left(\sqrt[3]{1-x}\right) \).

Step 2: Simplify the Logarithm of a Radical

Recall that the cube root of a number \( b \) can be written as \( b^{1/3} \). Therefore, \( \sqrt[3]{1-x} = (1-x)^{1/3} \). Substitute this into the expression: \( \log(x) - \log((1-x)^{1/3}) \).

Step 3: Apply the Power Rule for Logarithms

The power rule for logarithms states that \( \log(b^c) = c \cdot \log(b) \). Apply this rule: \( \log(x) - \frac{1}{3} \log(1-x) \).

Key Concepts

Quotient Rule for Logarithms

The Quotient Rule for Logarithms is an essential concept when dealing with expressions involving division inside a logarithm. It allows us to simplify such expressions by transforming the division into a subtraction of two separate logarithms. The rule is stated as follows:

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

This rule makes it easier to handle complex logarithmic expressions by breaking them down into simpler parts.
By applying the quotient rule, you can turn a potentially difficult expression into something more manageable. For example, if you start with an expression like \( \log \left( \frac{x}{\sqrt[3]{1-x}} \right) \), you can apply this rule to get \( \log(x) - \log(\sqrt[3]{1-x}) \). Now, each logarithm in this expression can be simplified separately, using additional logarithmic rules if necessary.
This process not only simplifies the expression but also aids in further calculations or transformations.

Power Rule for Logarithms

The Power Rule for Logarithms is a powerful tool when you encounter an exponent inside a logarithmic expression. The rule simplifies the calculation by allowing you to move the exponent in front of the logarithm as a multiplier. Here's how it works:

• \( \log(b^c) = c \cdot \log(b) \)

This transformation leverages the properties of exponents, making it easier to perform computations involving logarithms.
In practice, for an expression like \( \log((1-x)^{1/3}) \), the power rule allows us to express it as \( \frac{1}{3} \cdot \log(1-x) \).
Using the power rule can dramatically simplify expressions, especially in cases with fractional or high exponents. This approach is particularly beneficial in mathematical problem-solving as it aligns complex operations with fundamental properties of numbers, thus making the overall expression clearer and more intuitive.

Radicals and Exponents

Understanding radicals and exponents is critical when dealing with logarithms, particularly when conversions are involved. A radical like the cube root can always be rewritten in terms of an exponent. For example, a cube root, \( \sqrt[3]{a} \), is equivalent to \( a^{1/3} \). This conversion is key when applying logarithmic rules, as it transitions a radical into a form useable with the Power Rule.

• Radical form: \( \sqrt[3]{1-x} \)
• Exponential form: \( (1-x)^{1/3} \)

Understanding these conversions is useful in simplifying logarithmic expressions. By rewriting \( \sqrt[3]{1-x} \) as \( (1-x)^{1/3} \), it is clear how to proceed with logarithmic simplifications.
Working with logarithms often requires such conversions because they align expressions with established mathematical rules, allowing for straightforward application of logarithmic properties like the quotient and power rules. Mastering these concepts will lead to a better grasp of how logarithms can be expanded and simplified.