Problem 64 - College Algebra

Question

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ \frac{1}{3}\left(\log _{4} x-\log _{4} y\right) $$

Step-by-Step Solution

Answer

\(\log_4 \frac{x^{1/3}}{y^{1/3}}\)

Step 1: Utilize the Quotient Rule

The quotient rule states that the difference of two logs is the log of the quotient. So, apply this rule to the given expression: \(\frac{1}{3}(\log _{4} x-\log _{4} y) = \frac{1}{3} \log_4 \left(\frac{x}{y}\right)\).

Step 2: Apply the Power Rule

The power rule tells us that a coefficient in front of a log can be rewritten as an exponent inside the log. Thus, take the coefficient \(\frac{1}{3}\) inside to the argument of \(\log_4\), raising it to the power of \(\frac{1}{3}\): \( \frac{1}{3} \log_4 \left(\frac{x}{y}\right) = \log_4 \left(\frac{x}{y}\right)^{1/3}\).

Step 3: Redistribute the Power

By the power of a quotient rule, \((\frac{x}{y})^{1/3}\) can be expressed as \(\frac{x^{1/3}}{y^{1/3}}\). So, the final expression is: \(\log_4 \left(\frac{x}{y}\right)^{1/3} = \log_4 \frac{x^{1/3}}{y^{1/3}}\).

Key Concepts

Quotient Rule of Logarithms

When tackling complex logarithmic expressions, understanding the Quotient Rule of Logarithms is essential. The rule is straightforward: if you have two logarithms with the same base being subtracted, like \( \log_b(m) - \log_b(n) \), you can combine them into a single log by forming a quotient, so it becomes \( \log_b(\frac{m}{n}) \).

This method is particularly useful for simplifying expressions and solving logarithmic equations where the arguments are being divided. For example, if given the expression \( \log_4(x) - \log_4(y) \), we can apply the Quotient Rule to condense it into \( \log_4(\frac{x}{y}) \). It's a powerful tool that not only simplifies the expressions but also prepares them for further manipulation, such as applying the Power Rule or solving for variables.

Power Rule of Logarithms

Let's explore the Power Rule of Logarithms, which comes in handy when dealing with logarithmic expressions that involve exponents. In essence, the rule stipulates that a coefficient in front of the log can be converted into an exponent inside the logarithmic argument. This can be mathematically noted as \( \log_b(m^n) = n \cdot \log_b(m) \), and it also works in reverse.

When you have an expression like \( \frac{1}{3} \log_4(\frac{x}{y}) \), you can apply the Power Rule by 'moving' the coefficient into the argument to create \( \log_4(\left(\frac{x}{y}\right)^{1/3}) \). This step is indispensable for condensing logarithmic expressions and is indispensable for solving logarithmic equations elegantly. It also aligns with the objective of writing expressions with a coefficient of 1, as often required in mathematical exercises.

Logarithmic Expressions

Logarithmic expressions can sometimes seem intimidating, but by understanding the rules of logarithms, we can simplify even the most complex-looking expressions. To gain true mastery, it's necessary to become comfortable manipulating and transforming these expressions using various logarithmic properties, including the Quotient and Power Rules.

For instance, when you're presented with an expression such as \( \log_4 \left(\frac{x^{1/3}}{y^{1/3}}\right) \) after applying the rules, you've successfully condensed the expression into a more manageable form. This elegant simplified form not only helps in solving for variables but also in understanding the relationship between the components of a logarithmic expression. Keep in mind that evaluating when possible, such as when the argument is a known number, is a final step that can further simplify the expression.