Calculating 2log₆(33): A Step-by-Step Solution

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Hey guys! Today, we're diving into the world of logarithms to solve a pretty interesting problem. We need to calculate the value of the expression 2log₆(33) and round our answer to the third decimal place. Sounds a bit tricky, right? But don't worry, we'll break it down step by step so it's super easy to understand. Let's get started!

Understanding Logarithms

Before we jump into the calculation, let's quickly recap what logarithms are all about. Think of a logarithm as the inverse operation of exponentiation. In simpler terms, if we have an equation like b^x = y, the logarithm helps us find the exponent 'x'. We write this as log_b(y) = x. The 'b' here is the base of the logarithm, and 'y' is the argument.

So, what does log₆(33) mean? It's asking: "To what power must we raise 6 to get 33?" That's the key to understanding and solving logarithmic expressions.

Why are Logarithms Important?

Understanding logarithms is essential for various reasons. They pop up in many areas of math and science, including:

  • Solving Exponential Equations: Logarithms are the go-to tool when you need to solve for an unknown exponent.
  • Simplifying Complex Calculations: Logarithmic scales help us deal with very large or very small numbers, making them more manageable.
  • Modeling Real-World Phenomena: From measuring the intensity of earthquakes (the Richter scale) to calculating sound levels (decibels) and even in financial models, logarithms play a crucial role.

Step-by-Step Calculation of 2log₆(33)

Okay, now that we've refreshed our understanding of logarithms, let's tackle the problem at hand: calculating 2log₆(33). We'll go through it step by step.

Step 1: The Change of Base Formula

Most calculators can only compute logarithms with base 10 (log₁₀, often written simply as log) or base e (the natural logarithm, ln). Since our problem involves a logarithm with base 6, we'll need to use the change of base formula. This formula allows us to convert a logarithm from one base to another. Here’s the formula:

logₐ(b) = logₓ(b) / logₓ(a)

Where:

  • a is the original base (in our case, 6).
  • b is the argument (in our case, 33).
  • x is the new base (we can use either 10 or e).

For convenience, let's use base 10. So, we can rewrite log₆(33) as:

log₆(33) = log₁₀(33) / log₁₀(6)

Step 2: Calculate log₁₀(33) and log₁₀(6)

Now, we need to find the values of log₁₀(33) and log₁₀(6). Grab your calculator, and let's do it!

  • log₁₀(33) ≈ 1.5185
  • log₁₀(6) ≈ 0.7782

Step 3: Divide the Logarithms

Next, we divide the value of log₁₀(33) by the value of log₁₀(6):

log₆(33) ≈ 1.5185 / 0.7782 ≈ 1.9513

Step 4: Multiply by 2

Remember, the original expression was 2log₆(33). So, we need to multiply the result from Step 3 by 2:

2log₆(33) ≈ 2 * 1.9513 ≈ 3.9026

Step 5: Round to the Third Decimal Place

Finally, we need to round our answer to the third decimal place. Looking at 3.9026, the digit in the fourth decimal place is 6, which is greater than or equal to 5. So, we round up the third decimal place:

2log₆(33) ≈ 3.903

So, after all that calculating, we've found that the approximate value of 2log₆(33) rounded to the third decimal place is 3.903.

Why is Precision Important?

You might wonder, why bother with rounding to the third decimal place? In many real-world applications, precision is key. For example:

  • Engineering: Small errors in calculations can lead to significant problems in structural designs or electronic circuits.
  • Finance: Even tiny differences in interest rates or investment returns, when compounded over time, can result in substantial gains or losses.
  • Science: Accurate measurements are crucial for conducting experiments and validating scientific theories.

Rounding to a specific decimal place ensures that we maintain a certain level of accuracy in our results, which is particularly important when these results are used for further calculations or decision-making.

The Answer

Therefore, none of the provided options (a) 2.123, (b) 1.987, (c) 3.456, or (d) 2.678 correctly represent the calculated value. The correct answer, rounded to the third decimal place, is approximately 3.903.

Alternative Approaches and Tools

While we've walked through the manual calculation, there are alternative ways to find the value of 2log₆(33). Here are a couple of options:

Online Calculators

There are numerous online calculators available that can compute logarithms with any base. Simply search for a "logarithm calculator" and you'll find several options. These calculators allow you to directly input the base and argument, providing a quick and accurate result.

Scientific Calculators

Most scientific calculators have built-in functions for calculating logarithms with base 10 and base e (natural logarithms). Some advanced calculators can even handle logarithms with any base. Check your calculator's manual to see if it has this feature.

Programming Languages

If you're comfortable with programming, you can use languages like Python, R, or MATLAB to compute logarithms. These languages have built-in mathematical functions that make calculating logarithms a breeze. For example, in Python, you can use the math.log() function along with the change of base formula:

import math

result = 2 * (math.log(33, 6))
print(result)

Common Mistakes and How to Avoid Them

When working with logarithms, it's easy to make a few common mistakes. Here are some pitfalls to watch out for:

  • Forgetting the Order of Operations: Remember to follow the order of operations (PEMDAS/BODMAS). Logarithmic calculations should be done before multiplication or addition.
  • Incorrectly Applying the Change of Base Formula: Make sure you're using the change of base formula correctly. It's logₓ(b) / logₓ(a), not the other way around.
  • Misusing Calculator Functions: Double-check that you're using the correct logarithm function on your calculator (log₁₀, ln, or a general log with a specified base).
  • Rounding Errors: Be consistent with rounding. If you need to round to the third decimal place, do it at the final step to avoid accumulating errors.

Conclusion

So there you have it! We've successfully calculated the value of 2log₆(33) and rounded it to the third decimal place. We've also explored the importance of logarithms, walked through the step-by-step calculation, and discussed alternative approaches and common mistakes to avoid. Remember, practice makes perfect, so keep working on these types of problems, and you'll become a logarithm pro in no time!

I hope this breakdown was helpful, guys! If you have any questions or want to dive deeper into logarithms, just let me know. Keep exploring and happy calculating!