Equivalent Expression For Log₈ 4a((b-4)/c⁴) A Comprehensive Guide

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Understanding logarithms and their properties is crucial for simplifying complex expressions. In this article, we will delve into the properties of logarithms to determine the expression equivalent to log₈ 4a((b-4)/c⁴). We will break down the given expression step-by-step, applying logarithmic rules to arrive at the correct answer. This involves understanding the product rule, quotient rule, and power rule of logarithms. By the end of this article, you will not only know the solution but also have a comprehensive understanding of how to manipulate logarithmic expressions effectively.

Breaking Down the Logarithmic Expression

The given expression is log₈ 4a((b-4)/c⁴). To find the equivalent expression, we need to apply the properties of logarithms systematically. The primary rules we'll use are the product rule, quotient rule, and power rule.

1. Applying the Product Rule

The product rule of logarithms states that logₐ(xy) = logₐ(x) + logₐ(y). We can apply this rule to the expression inside the logarithm:

log₈ [4a((b-4)/c⁴)] = log₈(4a) + log₈((b-4)/c⁴)

Now, we apply the product rule again to log₈(4a):

log₈(4a) = log₈(4) + log₈(a)

So, our expression now looks like:

log₈(4) + log₈(a) + log₈((b-4)/c⁴)

2. Applying the Quotient Rule

The quotient rule of logarithms states that logₐ(x/y) = logₐ(x) - logₐ(y). Applying this rule to log₈((b-4)/c⁴), we get:

log₈((b-4)/c⁴) = log₈(b-4) - log₈(c⁴)

Now, our expression becomes:

log₈(4) + log₈(a) + log₈(b-4) - log₈(c⁴)

3. Applying the Power Rule

The power rule of logarithms states that logₐ(xⁿ) = n logₐ(x). We can apply this rule to -log₈(c⁴):

-log₈(c⁴) = -4 log₈(c)

Now, the complete expression is:

log₈(4) + log₈(a) + log₈(b-4) - 4 log₈(c)

Analyzing the Options

Now, let's compare our simplified expression with the given options:

A. log₈ 4 + log₈ a - log₈(b-4) - 4 log₈ c B. log₈ 4 + log₈ a + (log₈(b-4) - 4 log₈ c) C. log₈ 4a + log₈ b - 4 - 4 log₈ c - 4 D. log₈ 4

Our simplified expression is:

log₈(4) + log₈(a) + log₈(b-4) - 4 log₈(c)

Comparing this with the options, we find that option B matches our simplified expression:

log₈ 4 + log₈ a + (log₈(b-4) - 4 log₈ c)

Common Mistakes to Avoid

When working with logarithms, it's easy to make mistakes if you don't carefully apply the rules. Here are some common errors to watch out for:

  • Incorrect application of the product rule: For example, confusing logₐ(x + y) with logₐ(x) + logₐ(y). Remember, the product rule applies to the logarithm of a product, not a sum.
  • Misapplication of the quotient rule: Similar to the product rule, the quotient rule applies to the logarithm of a quotient, not a difference.
  • Forgetting the power rule: The power rule is crucial for simplifying expressions with exponents inside logarithms. Forgetting to apply this rule can lead to incorrect simplifications.
  • Errors in algebraic manipulation: Ensure you're correctly distributing signs and combining like terms when simplifying logarithmic expressions.

To avoid these mistakes, always double-check your steps and ensure you're applying the correct logarithmic properties.

Further Practice and Applications

To solidify your understanding of logarithmic properties, practice with a variety of expressions. Try simplifying expressions with different bases and complex combinations of products, quotients, and powers. You can also explore real-world applications of logarithms, such as in calculating pH levels, measuring earthquake magnitudes, and modeling exponential growth and decay.

Practice Problems

  1. Simplify: log₂(8x²/y³)
  2. Expand: log₅(25a³b/c²)
  3. Combine: 2 log₃(x) + log₃(y) - log₃(z)

Real-World Applications

  • pH Levels: The pH of a solution is calculated using logarithms, specifically pH = -log₁₀[H+], where [H+] is the concentration of hydrogen ions.
  • Earthquake Magnitudes: The Richter scale, used to measure the magnitude of earthquakes, is based on a logarithmic scale.
  • Exponential Growth and Decay: Logarithms are used to solve exponential growth and decay problems, such as population growth and radioactive decay.

By practicing and exploring these applications, you'll gain a deeper understanding of how logarithms are used in various fields.

Conclusion

In summary, we have successfully simplified the logarithmic expression log₈ 4a((b-4)/c⁴) by applying the product rule, quotient rule, and power rule of logarithms. The equivalent expression is log₈ 4 + log₈ a + (log₈(b-4) - 4 log₈ c), which corresponds to option B. Remember to practice applying these logarithmic properties to various expressions to enhance your understanding and skills. Understanding logarithms is essential not only for mathematics but also for numerous real-world applications.

By mastering these concepts, you will be well-equipped to tackle more complex logarithmic problems and appreciate their significance in various scientific and mathematical contexts. Keep practicing, and you'll find that logarithms become an intuitive and powerful tool in your mathematical arsenal. Understanding logarithms opens doors to solving a wide array of problems, from simple simplifications to complex scientific calculations.