Problem 83 - Algebra 2

Question

Expand each logarithm. \(\log _{4} \frac{\sqrt{x^{5} y^{7}}}{z w^{4}}\)

Step-by-Step Solution

Answer

The expanded form of \(\log _{4} \frac{\sqrt{x^{5} y^{7}}}{z w^{4}}\) is \(\frac{5}{2} \log _{4} x + \frac{7}{2} \log _{4} y - \log _{4} z - 4 \log _{4} w\).

Step 1: Split the logarithm of the fraction

We can write the logarithm of a fraction as the subtraction of the logarithm of the numerator from the logarithm of the denominator. So, \(\log _{4} \frac{\sqrt{x^{5} y^{7}}}{z w^{4}}\) becomes \(\log _{4} \sqrt{x^{5} y^{7}} - \log _{4} z w^{4}\).

Step 2: Split the logarithms of the products

The logarithm of a product can be written as the sum of the logarithms. So, each term can be further split as follows: \(\log _{4} \sqrt{x^{5} y^{7}} = \log _{4} x^{5/2} + \log _{4} y^{7/2}\) and \(\log _{4} z w^{4} = \log _{4} z +\log _{4} w^{4}\). Bringing everything together, we get the expression \((\log _{4} x^{5/2} + \log _{4} y^{7/2}) - (\log _{4} z +\log _{4} w^{4})\).

Step 3: Convert the logarithms with powers

The logarithm of a number to a certain power can be written as the power times the logarithm. Each of the terms can be written as: \(\log _{4} x^{5/2} = \frac{5}{2} \log _{4} x\), \(\log _{4} y^{7/2} = \frac{7}{2} \log _{4} y\), \(\log _{4} w^{4} = 4 \log _{4} w\). Our expression therefore, becomes \(\frac{5}{2} \log _{4} x + \frac{7}{2} \log _{4} y - \log _{4} z - 4 \log _{4} w\) as a final expanded expression.

Key Concepts

The Relationship Between Logarithms and Exponents

Logarithms and exponents are closely intertwined mathematical concepts. An understanding of this relationship can make dealing with logarithmic expressions much easier. In essence, a logarithm is the inverse operation of exponentiation. For example, if you have an equation like

• \(b^x = y\)

then the equivalent logarithmic form would be

• \(\log_b y = x\).

Logarithms ask the question, "to what power must the base be raised to get a specific number?" This interaction between exponents and logarithms is crucial in solving logarithmic problems, especially when expanding or simplifying logarithmic expressions. Understanding this relationship not only helps in mathematics but also in fields like computer science and physics, where logarithms are commonly used to model phenomena.

Understanding Logarithms of Fractions

When dealing with logarithms, one common type of problem involves fractions. The logarithm of a fraction can be expanded using a simple property of logarithms:

• \(\log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n\).

This property tells us that the logarithm of a fraction is equal to the difference between the logarithm of the numerator and the logarithm of the denominator. This principle is derived from the laws of exponents, where division corresponds to subtraction. For example, if you have the expression \(\log_4 \frac{\sqrt{x^5 y^7}}{zw^4}\), it can be decomposed into

• \(\log_4 \sqrt{x^5 y^7} - \log_4 zw^4\).

Decomposing logarithmic expressions using this fraction property simplifies complex problems and helps in breaking them down into manageable parts.

Exploring Logarithmic Expansion

Logarithmic expansion refers to expressing a complex logarithmic expression as a sum or difference of simpler terms. The key is to use logarithmic laws effectively:

• Product Rule: \(\log_b(mn) = \log_b m + \log_b n\)
• Quotient Rule: \(\log_b\left(\frac{m}{n}\right) = \log_b m - \log_b n\)
• Power Rule: \(\log_b(m^p) = p \log_b m\)

In practice, for example, take

• \(\log_4 \sqrt{x^5 y^7} = \log_4 x^{5/2} + \log_4 y^{7/2}\)
• and \(\log_4 zw^4 = \log_4 z + \log_4 w^4\),

you first expand using the product rule. Then you apply the power rule to further simplify these terms by moving exponents to the front, giving

• \(\frac{5}{2} \log_4 x + \frac{7}{2} \log_4 y - \log_4 z - 4 \log_4 w\).

Being comfortable with these rules makes logarithmic expressions much simpler and less intimidating for students. These expansions support computational ease across various scientific and engineering fields.