Problem 253 - Precalculus

Question

For the following exercises, expand each logarititim as much as possible. Rewite each expression as a sum, difference, or product of logs. $$ \log _{4}\left(\frac{\frac{x}{2}}{w}\right) $$

Step-by-Step Solution

Answer

\(\log_4(x) - \log_4(2) - \log_4(w)\)

Step 1: Apply the Quotient Rule

The logarithm of a quotient can be rewritten as the difference of the logarithms of the numerator and denominator. Therefore, apply the rule \[\log_b\left(\frac{A}{B}\right) = \log_b(A) - \log_b(B)\] on the given expression: \[\log_4\left(\frac{\frac{x}{2}}{w}\right) = \log_4\left(\frac{x}{2}\right) - \log_4(w)\] This helps separate the expression into two parts.

Step 2: Simplify the Numerator Using Quotient Rule

Now, notice that the numerator \(\frac{x}{2}\) can also be expressed in terms of logs. Again, apply the quotient rule of logarithms to further expand it: \[\log_4\left(\frac{x}{2}\right) = \log_4(x) - \log_4(2)\] Substitute this expression back into the equation from Step 1.

Step 3: Combine All Terms

Substituting the simplified expression from Step 2 back into the expression from Step 1, we have:\[\log_4(x) - \log_4(2) - \log_4(w)\]This expression cannot be simplified further in terms of basic logarithm rules.

Key Concepts

Quotient Rule of Logarithms

The Quotient Rule of logarithms is a key tool to help simplify and expand logarithmic expressions. This rule states that the logarithm of a division can be rewritten as the difference between the logarithms of the numbers being divided. In more mathematical terms, if you have an expression like \( \log_b \left(\frac{A}{B}\right) \), you can rewrite it as \( \log_b(A) - \log_b(B) \). This is incredibly useful when trying to break down complex expressions into simpler components.
By applying the Quotient Rule, you can take advantage of the properties of logarithms to make calculations more manageable. This rule is widely used in mathematics, especially when dealing with problems that involve logarithmic identities and simplifications.

• Start with a division inside a logarithm.
• Apply the Quotient Rule to separate the log into two parts: one for the numerator, one for the denominator.
• Each part of the division gets its own logarithm, allowing complex expressions to be more easily handled.

This method is particularly helpful in dealing with fractions inside a log, as seen in our example where \( \log_4 \left(\frac{\frac{x}{2}}{w}\right) \) simplifies to \( \log_4 \left(\frac{x}{2}\right) - \log_4(w) \). Using the Quotient Rule early on makes further expansion far easier.

Logarithm Properties

Logarithm properties are a collection of rules that make it easier to manipulate and simplify logarithmic expressions. Understanding these properties is crucial for both algebraic manipulation and solving real-world problems involving logarithms. Some of the primary properties include:

• Product Rule: \( \log_b(AB) = \log_b(A) + \log_b(B) \)
• Quotient Rule: \( \log_b \left(\frac{A}{B}\right) = \log_b(A) - \log_b(B) \)
• Power Rule: \( \log_b(A^n) = n \cdot \log_b(A) \)
• Change of Base Formula: \( \log_b(A) = \frac{\log_c(A)}{\log_c(b)} \)

These properties help in breaking down complex logarithmic expressions into simpler components.
For example, knowing that the logarithm of a fraction can be expressed as a difference (Quotient Rule), or how a power inside the log can be brought out front as a multiplier (Power Rule), allows for easier simplification. In our particular example, we first utilized the Quotient Rule to break down \( \log_4 \left(\frac{x}{2}\right) \) into \( \log_4(x) - \log_4(2) \). These structural tools provided by logarithm properties streamline the process of converting lengthy expressions into more manageable forms, aiding both understanding and calculation.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions often requires a logical application of logarithmic rules and properties. The main goal is to break down a given logarithmic expression into simpler and more digestible parts, making them easier to interpret or compute. This involves repeatedly applying relevant logarithmic rules, like the product, quotient, and power rules, to dissect the expression into its simplest form.
Start by identifying any fractions, products, or powers within the logarithm. Each of these can often be expanded into separate terms thanks to logarithmic properties. In our example, we took \( \log_4 \left(\frac{\frac{x}{2}}{w}\right) \) and simplified it step by step:

• First, applied the quotient rule to separate the complicated fraction.
• Next, broke the resulting expression \( \log_4 \left(\frac{x}{2}\right) \) into yet another simpler expression.
• Ended with \( \log_4(x) - \log_4(2) - \log_4(w) \).

This last form is the most expanded version possible under basic logarithm rules, as it expresses each element of the original fraction separately. Efficient simplification makes further analysis or solving much clearer and quicker, enabling more effective problem-solving in both academic and practical scenarios.