Problem 32 - College Algebra

Question

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \operatorname{og} \sqrt[5]{\frac{x}{y}} $$

Step-by-Step Solution

Answer

The expanded form of \(log\sqrt[5]{\frac{x}{y}}\) is \(\frac{1}{5}log(x) - \frac{1}{5}log(y)\).

Step 1: Break down the root of the logarithm

We know that the nth root of a number can be written as that number to the power of \(1/n\). Using this property, we can rewrite the expression inside the logarithm as follows: \(log((x/y)^{1/5})\).

Step 2: Apply the power rule of logarithms

The power rule of logarithms states that log(b^p) = plog(b). We apply this rule to the expression obtained from step 1: \(1/5 * log(x/y)\).

Step 3: Apply the quotient rule of logarithms

The quotient rule of logarithms states that log(a/b) = log(a) - log(b). Applying this to the expression from step 2, we get: \(1/5 * (log(x) - log(y))\). Thus we have successfully expanded the given expression.

Step 4: Final expansion

The final step is to expand our above result as follows: \(1/5 * log(x) - 1/5 * log(y)\).

Key Concepts

Properties of Logarithms

Understanding the basic properties of logarithms is crucial for expanding and simplifying complex logarithmic expressions. Here are some important properties to keep in mind:

• Product Rule: This rule states that the logarithm of a product is the sum of the logarithms of the factors. In mathematical terms, \( \log_b(MN) = \log_b(M) + \log_b(N) \).


• Quotient Rule: This rule states that the logarithm of a quotient is the difference between the logarithm of the numerator and the logarithm of the denominator. It's represented as \( \log_b(M/N) = \log_b(M) - \log_b(N) \).


• Power Rule: The power rule tells us that the logarithm of a power can be rewritten as the exponent times the logarithm of the base. Expressed as \( \log_b(M^p) = p \cdot \log_b(M) \).


• Change of Base Formula: Although not directly used in expanding logarithms, this is a useful property that allows us to change the base of a logarithm: \( \log_b(M) = \frac{\log_k(M)}{\log_k(b)} \), where \(k\) is any positive number.

Mastering these properties will provide you with the tools needed to tackle any logarithmic expression.

Power Rule of Logarithms

The power rule of logarithms is an essential tool when dealing with exponents in logarithmic expressions. According to the power rule:

• When you have an expression like \( \log_b(M^p) \), it can be rewritten by bringing the exponent \(p\) to the front as a multiplier: \( p \cdot \log_b(M) \).

This rule is particularly helpful in simplifying expressions and, as seen in our example, is useful when dealing with fractional exponents such as roots. Rewriting the expression \( \log((x/y)^{1/5}) \) as \( \frac{1}{5} \cdot \log(x/y) \) showcases the application of the power rule, making the expression much easier to handle in further steps. It's a straightforward method that can significantly simplify the complexity of logarithms.

Quotient Rule of Logarithms

The quotient rule of logarithms is used to split a single logarithmic expression of a division into two separate terms. The rule can be stated as:

• For any expression in the form of \( \log_b(M/N) \), it can be expanded into \( \log_b(M) - \log_b(N) \).

In our example, after applying the power rule, the expression becomes \( \frac{1}{5} \cdot \log(x/y) \). Here, the quotient rule allows us to split this into two simpler terms: \( \frac{1}{5} \cdot \log(x) - \frac{1}{5} \cdot \log(y) \). This transformation simplifies the expression and makes it easier to interpret or further manipulate. Recognizing when and how to apply the quotient rule allows for efficient solving and expression expansion.

Expanding Logarithmic Expressions

Expanding logarithmic expressions involves using the properties of logarithms to break down complex expressions into simpler, more manageable parts. The overall process can be summarized in a few steps:

• Identify the Structure: Look at the expression to see if it involves any products, quotients, or powers that can be separated using the logarithm rules.


• Apply Logarithm Rules: Use the product, quotient, and power rules to expand the expression. For example, the expression \( \log\sqrt[5]{\frac{x}{y}} \) can first be rewritten as \( \log((x/y)^{1/5}) \), then further expanded by applying these rules.


• Simplify the Result: Once expanded, simplify the expression by performing any multiplication or subtraction specified. This leads to a clean, expanded form that is both easier to handle and interpret.

Expanding expressions like \( \log((x/y)^{1/5}) \) into \( \frac{1}{5} \cdot \log(x) - \frac{1}{5} \cdot \log(y) \) demonstrates the power of these rules, turning a complex expression into a straightforward one. This method is not only useful for solving problems but also for deepening one's understanding of logarithmic functions.