Problem 11 - Algebra and Trigonometry

Question

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{4}\left(\frac{64}{y}\right) $$

Step-by-Step Solution

Answer

The expanded form of the expression \(\log_{4}\left(\frac{64}{y}\right)\) is \(3 - \log_{4}(y)\).

Step 1: Applying Quotient Rule

First, we are going to apply the quotient rule of logarithms, which states that \(\log_b(M/N) = \log_b(M) - \log_b(N)\).

Step 2: Applying the rule

Applying the rule to our expression, we get \(\log_{4}(64) - \log_{4}(y)\).

Step 3: Applying Power Rule

Next, since we know that \(4^3 = 64\), we can express \(\log_{4}(64)\) as 3 using the power rule of logarithms that states \(\log_b(b^x) = x\). So the expression now becomes \(3 - \log_{4}(y)\).

Key Concepts

Quotient Rule

The quotient rule is an essential property of logarithms that allows us to express the logarithm of a quotient as the difference of two separate logarithms. In simpler terms, if you have a log expression where two numbers are divided, you can rewrite it more simply as a subtraction:

• Formula: \(\log_b(M/N) = \log_b(M) - \log_b(N)\)

This rule becomes handy when simplifying complex logarithmic expressions. For example, in our exercise, the expression \[ \log_4\left(\frac{64}{y}\right) \]can be rewritten using the quotient rule as:

• \(\log_{4}(64) - \log_{4}(y)\)

This makes the expression clean and manageable. Every time you see a division inside a logarithm, think of the quotient rule as your tool for simplification.

Power Rule

The power rule of logarithms is a useful property that allows you to simplify expressions involving exponential forms inside a logarithm. It states that if you have a logarithm of a power, you can bring the exponent out in front of the logarithm:

• Formula: \(\log_b(b^x) = x \)

When dealing with logarithmic expressions that include a base raised to a power, the power rule can transform your expression into something much more manageable. Consider the part of our exercise, \( \log_4(64) \). Given that \( 4^3 = 64 \), by applying the power rule, we can easily simplify \( \log_4(64) \) to 3. This is because the logarithm base matches the base of the exponent, turning what might be a complex calculation into a basic arithmetic operation.

Logarithmic Expressions

Logarithmic expressions represent calculations involving logarithms, a mathematical function that helps us in handling exponential relationships. In our original exercise, we were given a single logarithmic expression: \[ \log_4\left(\frac{64}{y}\right) \]. To make sense of such expressions, we typically aim to simplify them using properties of logarithms.
A key strategy when working with logarithmic expressions is to identify potential applications of logarithmic rules such as the quotient, product, and power rules. This is because a complex expression can often be broken down into simpler parts, which can be evaluated individually without a calculator.
The main goal in expanding a logarithmic expression is to rewrite it in a form where it can be more easily interpreted or calculated. This process involves:

• Breaking expressions into simpler components using logarithmic properties.
• Identifying constants that can be calculated directly (like \(\log_4(64) \) turning into \(3\)).
• Making sure each part of the expression is as straightforward as possible.

Therefore, by leveraging these expression rules, we can tackle seemingly difficult problems in an intuitive and logical way.

Logarithm Base Conversion

Converting a logarithmic expression to a different base can sometimes simplify problems or help us better compare different logarithms. The base of a logarithm is essentially the number we repeatedly multiply. Sometimes, it’s necessary to convert a logarithm to another base to simplify multiplication or addition.Even though the provided exercise did not require converting logarithms to a different base, it is still a valuable skill. The conversion formula is as follows:

• \(\log_b(a) = \frac{\log_k(a)}{\log_k(b)}\)

This allows you to change the base of any logarithm to any other base \(k\), including commonly used bases such as 10 or \(e\) (natural logarithm). This conversion can help when evaluating logarithms with unfamiliar bases or when using a calculator that is optimized for base 10 or \(e\).
Understanding how to navigate between different bases using this conversion rule enhances your ability to solve a wider variety of logarithmic problems, whether they appear in exams or real-life applications.