Problem 82 - Algebra and Trigonometry

Question

Expand: \(\log _{8}\left(\frac{\sqrt[4]{x}}{64 y^{3}}\right) .\) (Section 4.3, Example 4)

Step-by-Step Solution

Answer

The expanded form of the expression is \(\frac{1}{4} \log_8 (x) - 2 - 3 \log_8 (y)\).

Step 1: Apply the Logarithm Division Rule

Start by applying the rule for division inside the logarithm: \(\log_b(m/n) = \log_bm - \log_bn\). This rule transforms \(\log _{8}\left(\frac{\sqrt[4]{x}}{64 y^{3}}\right)\) into a subtraction: \(\log_8 (\sqrt[4]{x}) - \log_8 (64 y^{3})\).

Step 2: Apply the Logarithm Product Rule on the Second Term

Now apply the rule for multiplication inside the logarithm for the second part of the subtracted fraction. The rule \(\log_b(mn) = \log_bm + \log_bn\) transforms \(\log_8 (64 y^{3})\) into a sum: \(\log_8 (64) + \log_8 (y^3)\). So, the entire expression now becomes: \(\log_8 (\sqrt[4]{x}) - (\log_8 (64) + \log_8 (y^3))\). Since \(\log_8 64 = 2\) (because \(8^2 = 64\)), the expression simplifies to: \(\log_8 (\sqrt[4]{x}) - 2 - \log_8 (y^3)\).

Step 3: Apply the Power Rule for Logarithms

Finally, apply the power rule \(\log_b(m^n) = n \cdot \log_bm\) which allows us to bring the exponent in front of the logarithm. So, \(\log_8 (\sqrt[4]{x})\) becomes \(\frac{1}{4} \log_8 (x)\) and \(\log_8 (y^3)\) becomes \(3 \log_8 (y)\). The entire expression now simplifies to: \(\frac{1}{4} \log_8 (x) - 2 - 3 \log_8 (y)\).

Key Concepts

Logarithm Division Rule

The logarithm division rule is a fundamental property that helps in simplifying logarithmic expressions involving division. When you have a logarithm of a division, such as \( \log_b\left(\frac{m}{n}\right) \), you can split it into the difference of two logarithms, which will be \( \log_b(m) - \log_b(n) \).

This property is incredibly practical because it enables you to handle complicated expressions by breaking them down into more manageable parts. In our exercise, applying this rule converted a single logarithmic term with a fraction inside into a difference of two separate logs, simplifying the expression significantly.

Logarithm Product Rule

The logarithm product rule is an equally important property to understand. It states that the logarithm of a product is the sum of the logarithms of the individual factors: \( \log_b(mn) = \log_b(m) + \log_b(n) \).

This property allows you to deconstruct a single log with a multiple term inside into a sum of logs, each with a single term. For instance, in our exercise, \( \log_8 (64 y^3) \) was transformed into \( \log_8 (64) + \log_8 (y^3) \), significantly simplifying the equation. Understanding this property is crucial for working with logarithmic expressions involving multiplication.

Power Rule for Logarithms

When dealing with logarithms of exponential terms, the power rule for logarithms comes into play. It states that the logarithm of a number raised to an exponent is equal to the exponent times the logarithm of the number: \( \log_b(m^n) = n \cdot \log_b(m) \).

This rule is incredibly useful because it allows you to move the exponent to the front of the logarithm, turning an exponential expression into a multiplication, which is often easier to work with. In the example given, the power rule helped us rewrite \( \log_8 (\sqrt[4]{x}) \) as \( \frac{1}{4} \log_8 (x) \) and \( \log_8 (y^3) \) as \( 3 \log_8 (y) \), thus simplifying the expression.

Simplifying Logarithmic Expressions

To simplify logarithmic expressions, you often need to apply a combination of logarithmic rules. It begins by understanding each individual property and then skillfully applying them to break down complex terms into simpler ones.

In our exercise, we successfully simplified \( \log _{8}\left(\frac{\sqrt[4]{x}}{64 y^{3}}\right) \) by applying all these rules step by step, ultimately reducing it to \( \frac{1}{4} \log_8 (x) - 2 - 3 \log_8 (y) \). This strategy of simplification not only makes the expressions easier to work with but also paves the way for further algebraic manipulation and solving of equations.