Question

If \(\log _{4} \frac{x^{4}}{4}+3 \log _{4} 4 x^{4}=p+q \log _{4} x\), then the value of \(\log _{4}(q)\) is _______. (1) 4 (2) \(-4\) (3) 3 (4) 2

Short answer

Based on the step by step solution provided above, the correct answer is: The value of \(\log _{4}(q)\) is \(3+28\log _{4} x\).

Step-by-step solution

Simplify the given logarithmic expression

We have the given equation: \(\log _{4} \frac{x^{4}}{4}+3 \log _{4} 4 x^{4}=p+q \log _{4} x\) We'll start by simplifying the left side of the equation using the properties of logarithms.

Use the quotient rule

The quotient rule states that \(\log_b (\frac{a}{b}) = \log_b a - \log_b b\). Applying this rule to the first term on the left side, we get: \(\log _{4} x^{4} - \log_4 4+3 \log _{4} 4 x^{4}=p+q \log _{4} x\)

Apply the log rule for powers

The log rule for powers states that \(n \log_b a = \log_b a^n\). Apply this rule to both the first and third term: \(\log _{4} x^{16} - \log_4 4 + \log _{4} (4 x^4)^3 = p + q \log _{4} x\)

Simplify the log base terms

As \(\log_4 4 =1\), the equation will be as follows: \(\log _{4} x^{16} - 1 + \log _{4} (64 x^{12}) = p + q \log _{4} x\) Now, the left side consists of two logarithmic terms and a constant.

Combine the log terms

Using the product rule of logarithm, \(\log_b A + \log_b B = \log_b(AB)\), on the log terms on the left side, we get: \(\log_{4}(x^{16} \cdot 64 x^{12}) - 1 = p + q \log_{4} x\)

Simplify the expression inside the log

Multiply the similar parts inside the log and simplify: \(\log_{4}(64 x^{28}) - 1 = p + q \log_{4} x\) Now, comparing this equation with the original equation: \(\log_{4}(64 x^{28}) - 1 = p + q \log_{4} x\) It is clear that \(p=-1\) and \(q=64x^{28}\).

Find the value of \(\log_{4}(q)\)

Substitute the value of \(q\) found in step 6: \(\log_{4}(64 x^{28})\) Since \(64=4^3\), we can rewrite this as: \(\log_{4}(4^3 x^{28})\) Now, applying the rules of logarithm: \(3 + 28 \log_{4} x\) The answer will be in the form of \(3+28\log_{4} x\). From the given options, the correct answer is: \(\boxed{(3)}\)

Key concepts

Properties of Logarithms

Understanding logarithms can be simplified by learning their properties, which are crucial tools in transforming complex expressions. One key property is the **product rule**: \(\log_b (mn) = \log_b m + \log_b n\). This allows us to break apart or combine logarithms of multiplied terms. Another important rule is the **quotient rule**: \(\log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n\), which helps separate or consolidate terms within a division. Finally, we have the **power rule**: \(\log_b (m^n) = n \log_b m\). This property allows us to move exponents outside of the logarithm, making simplification easier. Together, these properties provide a powerful way to handle and manipulate logarithmic expressions.

Simplifying Logarithms

When given a complex logarithmic expression, breaking it down into simpler parts is often a crucial step. In the exercise, this involves using the properties of logarithms effectively. To simplify the given equation, apply the **quotient rule** to the term \(\log_4 \frac{x^4}{4}\) to split it into two separate terms. Then, use the **power rule** on expressions like \(3 \log_4 4x^4\) to deal with the powers directly. Through these steps, you can transform the expression into more manageable components, ultimately allowing for straightforward comparison with other parts of the problem.

Logarithmic Rules

Logarithmic equations rely heavily on certain rules, which serve as the tools needed to solve them efficiently and accurately. The most pertinent include the product, quotient, and power rules, but it's also critical to recognize the concept of changing bases or simplifying expressions like \(\log_4 4 = 1\). Recognizing relations such as \(\log_b b^x = x\) can sometimes allow direct calculations without further simplification. By consistently applying these rules, seemingly complex logarithmic problems become much more approachable and understandable.

Exponential Expressions

Logarithms and exponents are inherently linked because logarithms are essentially inverses of exponentials. In exercises like the one described, recognizing exponential relationships can enable simplification. For instance, realizing that \(64 = 4^3\) allows us to express a component as a power of the logarithm's base. This, in turn, lets us apply logarithmic rules more effectively. Exponential thinking helps condense expressions; reducing \(4^3 x^{28}\) gives you terms amenable to power rules, converting difficult operations into simpler, more solvable forms. Understanding this relationship is key to managing both logarithmic and exponential expressions effectively.