Unmasking Logarithmic Errors A Step By Step Analysis

Hey there, math enthusiasts! Today, we're diving deep into the fascinating world of logarithms and dissecting a common pitfall that many students encounter. We'll be analyzing Marte's attempt to simplify a logarithmic expression, pinpointing the exact step where she went astray. So, buckle up, grab your thinking caps, and let's embark on this mathematical adventure!

The Logarithmic Labyrinth Marte's Journey

Our central mathematics problem revolves around Marte's simplification of the expression:

$$4 \log _5 x+\log _5 2 x-\log _5 3 x$$

Marte's journey through this mathematical labyrinth led her to the following steps:

Step 1: $ \log _5 4 x+\log _5 2 x-\log _5 3 x$

Step 2: $ \log _5(4 x+2 x)-\log _5 3 x$

Step 3: $ \log _5 6 x-\log _5 3 x$

Step 4: $ \log _5 \frac{6 x}{3 x}$

Step 5: $ \log _5 2$

Our mission, should we choose to accept it, is to identify the precise moment where Marte took a wrong turn. We'll meticulously examine each step, armed with the fundamental properties of logarithms, to unmask the culprit.

Dissecting the Steps The Devil is in the Details

To truly understand where Marte faltered, we need to dissect each step with the precision of a surgeon. Let's equip ourselves with the essential properties of logarithms that will serve as our guiding principles.

Key Logarithmic Properties

• Power Rule: $\log_b (x^p) = p \log_b (x)$ This rule allows us to move exponents inside a logarithm to the front as a coefficient, and vice versa.
• Product Rule: $\log_b (x) + \log_b (y) = \log_b (xy)$ When adding logarithms with the same base, we can combine them into a single logarithm by multiplying their arguments.
• Quotient Rule: $\log_b (x) - \log_b (y) = \log_b (\frac{x}{y})$ When subtracting logarithms with the same base, we can combine them into a single logarithm by dividing their arguments.

With these powerful tools in our arsenal, let's scrutinize Marte's steps.

Step 1 Unveiling the Power Rule

In Step 1, Marte transforms the initial expression from $4 \log _5 x+\log _5 2 x-\log _5 3 x$ to $\log _5 4 x+\log _5 2 x-\log _5 3 x$. This is where the crucial power rule comes into play. The power rule of logarithms, which states that $\log_b (x^p) = p \log_b (x)$, was not applied correctly here. Instead of bringing the coefficient 4 inside the logarithm as an exponent, Marte seems to have simply multiplied it with the argument $x$, which is a critical error.

The correct application of the power rule would involve recognizing that $4 \log_5 x$ is equivalent to $\log_5 (x^4)$. This subtle yet significant difference changes the entire course of the simplification. This misapplication of the power rule is the first misstep in Marte's calculation.

To solidify this understanding, let's consider a numerical example. Suppose $x = 2$. Then, $4 \log_5 2$ is approximately $4 * 0.4307 = 1.7228$, while $\log_5 (4 * 2) = \log_5 8$ is approximately 1.2920. Clearly, these values are not equal, highlighting the error in Marte's approach. This mathematical misstep leads to an incorrect simplification.

Step 2 The Addition Illusion

Step 2 presents another mathematical challenge. Marte combines $ \log _5 4 x+\log _5 2 x$ into $ \log _5(4 x+2 x)$. Here, she attempts to apply the properties of logarithms, but in an incorrect manner. The product rule of logarithms states that $\log_b (x) + \log_b (y) = \log_b (xy)$. This means that the sum of two logarithms with the same base is equal to the logarithm of the product of their arguments, not the sum. Marte's misunderstanding of this rule leads to an incorrect combination of the logarithmic terms.

The correct application of the product rule would transform $ \log _5 4 x+\log _5 2 x$ into $ \log _5(4x * 2x) = \log_5 (8x^2)$. This crucial distinction highlights the error in Marte's calculation. Adding the arguments directly, as Marte did, is a fundamental violation of logarithmic principles.

To illustrate this point, let's consider a simple example. Suppose $x = 2$. Then, $\log_5 (4 * 2) + \log_5 (2 * 2) = \log_5 8 + \log_5 4$ which is approximately $1.2920 + 0.8614 = 2.1534$. However, $\log_5 (4 * 2 + 2 * 2) = \log_5 12$ is approximately 1.5440. These different results underscore the incorrectness of adding the arguments directly.

Step 3 Simplifying Incorrectly

Step 3, which simplifies $ \log _5(4 x+2 x)-\log _5 3 x$ to $ \log _5 6 x-\log _5 3 x$, is a direct consequence of the error made in Step 2. Since the previous step was incorrect, this step inherits that error. While the arithmetic within the argument is correct (4x + 2x does indeed equal 6x), the initial mistake of adding arguments within logarithms makes this entire step invalid.

Step 4 The Quotient Rule Rescue (Almost)

In Step 4, Marte applies the quotient rule correctly, transforming $ \log _5 6 x-\log _5 3 x$ into $ \log _5 \frac{6 x}{3 x}$. The quotient rule, which states that $\log_b (x) - \log_b (y) = \log_b (\frac{x}{y})$, is applied accurately here. However, this correct application is built upon the foundation of previous errors, rendering the result still incorrect.

Step 5 The Final (Incorrect) Answer

Step 5 completes the simplification, reducing $ \log _5 \frac{6 x}{3 x}$ to $ \log _5 2$. The simplification of the fraction within the logarithm is correct (6x / 3x = 2). However, due to the accumulated errors from earlier steps, this final answer is not the correct simplification of the original expression. It’s a classic case of “garbage in, garbage out” – even a correct step can't salvage an incorrect process.

The Real Culprit Step 1's Logarithmic Lapse

After our detailed investigation, the culprit is clear: Step 1 is where Marte incorrectly applied a property of logarithms. By failing to correctly apply the power rule, she set off a chain reaction of errors that led to an incorrect final answer. Specifically, instead of recognizing that $4 \log_5 x$ should be transformed into $\log_5 (x^4)$, she incorrectly treated it as $\log_5 (4x)$. This seemingly small mistake had significant consequences.

To reiterate, the error in Step 1 is the incorrect application of the power rule of logarithms. This rule states that $\log_b (x^p) = p \log_b (x)$. Marte should have transformed $4 \log_5 x$ into $\log_5 (x^4)$ before proceeding with any further simplification. Her failure to do so led to an incorrect result.

Correcting Marte's Course The Right Path

Now that we've identified the error, let's set Marte back on the right path. We'll correctly simplify the expression, demonstrating the proper application of logarithmic properties.

Starting with the original expression:

$$4 \log _5 x+\log _5 2 x-\log _5 3 x$$

1. Apply the Power Rule: We correctly apply the power rule to the first term:

   $$\log _5 x^4+\log _5 2 x-\log _5 3 x$$

2. Apply the Product Rule: Next, we use the product rule to combine the first two logarithms:

   $$\log _5 (x^4 * 2 x)-\log _5 3 x$$

   $$\log _5 (2 x^5)-\log _5 3 x$$

3. Apply the Quotient Rule: Now, we apply the quotient rule to combine the remaining logarithms:

   $$\log _5 \frac{2 x^5}{3 x}$$

4. Simplify: Finally, we simplify the expression within the logarithm:

   $$\log _5 \frac{2 x^4}{3}$$

Therefore, the correct simplification of the expression is:

$$\log _5 \frac{2 x^4}{3}$$

This correct solution stands in stark contrast to Marte's final answer, highlighting the importance of meticulous application of logarithmic properties.

Lessons Learned Avoiding Logarithmic Lapses

Marte's mathematical journey, though fraught with errors, provides valuable lessons for all students of mathematics. The key takeaways are:

1. Master the Properties: A deep understanding of logarithmic properties is paramount. The power rule, product rule, and quotient rule are your allies in the world of logarithms. Know them, love them, and apply them correctly.
2. Step-by-Step Precision: Each step in a simplification process must be meticulously executed. Avoid mental shortcuts and write out each step clearly. This minimizes the risk of errors.
3. Double-Check Your Work: After each step, take a moment to review your work. Does it align with the properties of logarithms? Are there any potential errors? A little extra vigilance can save you from significant mistakes.
4. Practice Makes Perfect: Like any mathematical skill, proficiency in simplifying logarithmic expressions comes with practice. Work through a variety of problems, and don't be afraid to make mistakes – they are valuable learning opportunities.

Final Thoughts The Power of Precision

In conclusion, Marte's error serves as a powerful reminder of the importance of precision in mathematics. A single mistake, particularly in the early stages of a problem, can cascade through the entire solution, leading to an incorrect final answer. By mastering the properties of logarithms, practicing diligently, and approaching each step with meticulous care, we can avoid logarithmic lapses and conquer the mathematical world!

So, guys, let’s remember this journey and strive for accuracy in all our mathematical endeavors. Keep those logarithmic properties close, and happy simplifying!