Rewriting Log(1/100) Using The Power Rule Of Logarithms

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Hey guys! Today, we're diving into the fascinating world of logarithms and how we can manipulate them using some cool rules. Specifically, we're going to break down how to rewrite the expression log(1/100) using the power rule of logarithms. This rule is a game-changer when dealing with exponents inside logarithms, so let's get started!

Understanding the Power Rule of Logarithms

So, what exactly is this power rule we keep mentioning? Well, in simple terms, the power rule of logarithms states that the logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number. Mathematically, it looks like this:

logb(xk) = k * logb(x)

Where:

  • b is the base of the logarithm (usually 10 if not specified, known as the common logarithm).
  • x is the number we're taking the logarithm of.
  • k is the exponent or power.

This rule is super handy because it allows us to bring exponents outside the logarithm, making complex expressions much easier to handle. Think of it as a way to simplify logarithmic calculations and make them more manageable. Why is this important? Well, in many real-world applications, especially in fields like physics, engineering, and finance, you'll encounter situations where you need to solve equations involving logarithms. The power rule helps you isolate variables and solve for unknowns efficiently.

For example, imagine you're working on a problem involving compound interest, where the amount of money you have after a certain time is given by a logarithmic equation. By using the power rule, you can bring the exponent (which might represent the time period) outside the logarithm, making it easier to solve for the interest rate or the principal amount. Similarly, in physics, you might encounter logarithmic scales like the Richter scale for earthquake magnitude or the decibel scale for sound intensity. The power rule can be crucial in converting between these scales and understanding the relative magnitudes of the phenomena being measured. In essence, mastering the power rule of logarithms opens up a whole new world of problem-solving capabilities in various fields. So, let’s see how we can apply this rule to our specific problem of rewriting log(1/100).

Rewriting log(1/100) Using the Power Rule

Now, let's get to the core of the question: How do we rewrite log(1/100) using the power rule? The first step is to recognize that 1/100 can be expressed as a power of 10. Specifically, 1/100 is equal to 10-2. This is a crucial transformation because it allows us to directly apply the power rule. Remember, negative exponents indicate reciprocals, so 10-2 means 1/(102), which is 1/100. So, we can rewrite our expression as:

log(1/100) = log(10-2)

Now, the power rule of logarithms comes into play. We can take the exponent (-2) and bring it down as a coefficient, multiplying it by the logarithm of the base (10). This gives us:

log(10-2) = -2 * log(10)

This is where the simplification truly happens. We've successfully used the power rule to rewrite the expression. But we can go one step further. What is log(10)? Remember that when we write "log" without a base, we're usually referring to the common logarithm, which has a base of 10. So, log(10) is asking, "To what power must we raise 10 to get 10?" The answer, of course, is 1.

Therefore, log(10) = 1. Substituting this back into our expression, we get:

-2 * log(10) = -2 * 1 = -2

So, rewriting log(1/100) using the power rule of logarithms gives us -2 * log(10), which simplifies to -2. This demonstrates the power and efficiency of the power rule in simplifying logarithmic expressions. This process not only provides the answer but also reinforces the understanding of how logarithmic functions work and how the power rule can be a powerful tool in solving mathematical problems.

Analyzing the Answer Choices

Alright, now that we've successfully rewritten log(1/100) and arrived at the simplified form of -2 * log(10), let's take a look at the answer choices and see which one matches our result. This is a crucial step in problem-solving, as it ensures we haven't made any errors along the way and helps us select the correct answer.

The provided answer choices are:

A. 2 log -10 B. 10 log -2 C. -2 log 10 D. log -20

Let's analyze each option:

  • A. 2 log -10: This option is incorrect for a couple of reasons. First, the argument of a logarithm (the value inside the logarithm) cannot be negative. Logarithms are only defined for positive numbers. Second, even if we ignored the negative sign, this option represents 2 multiplied by the logarithm of 10, not -2 multiplied by the logarithm of 10.
  • B. 10 log -2: Similar to option A, this choice is also incorrect because it involves the logarithm of a negative number (-2), which is undefined. Additionally, the structure of this expression doesn't match our simplified form of -2 * log(10).
  • C. -2 log 10: This option perfectly matches our result! We derived that log(1/100) can be rewritten as -2 * log(10) using the power rule of logarithms. This is the correct answer.
  • D. log -20: This option is incorrect for the same reason as options A and B: it involves the logarithm of a negative number (-20), which is not defined.

Therefore, the correct answer is C. -2 log 10. This process of analyzing the answer choices not only confirms our solution but also reinforces our understanding of the properties of logarithms and how to correctly apply them. By eliminating incorrect options based on fundamental logarithmic principles, we can confidently arrive at the correct answer and enhance our problem-solving skills.

Why is this Important?

You might be thinking, "Okay, that's cool, but why do I even need to know this?" Well, the power rule of logarithms isn't just some abstract mathematical concept. It's a powerful tool that has real-world applications in various fields.

  • Science and Engineering: Logarithms are used extensively in fields like physics, chemistry, and engineering. For example, the Richter scale, which measures the magnitude of earthquakes, is a logarithmic scale. The decibel scale, used to measure sound intensity, is also logarithmic. The power rule can help you compare and analyze these measurements.
  • Finance: Logarithms are used in financial calculations, such as compound interest. Understanding the power rule can help you calculate investment growth and understand financial models.
  • Computer Science: Logarithms are used in algorithm analysis. They help determine the efficiency of algorithms and how they scale with increasing data size.
  • Everyday Life: Even in everyday life, understanding logarithms can be helpful. For example, understanding exponential growth, which is closely related to logarithms, can help you make informed decisions about things like investments and population growth.

In essence, the power rule of logarithms is a fundamental concept that provides a foundation for understanding and solving problems in a wide range of disciplines. By mastering this rule, you're equipping yourself with a valuable tool that can help you in your academic pursuits, your career, and even your daily life. So, the next time you encounter a logarithmic expression, remember the power rule and how it can simplify your calculations and provide valuable insights.

Conclusion

So, there you have it! We've successfully rewritten log(1/100) using the power rule of logarithms, step by step. We saw how expressing 1/100 as 10-2 allowed us to apply the rule and simplify the expression. By bringing the exponent down as a coefficient, we transformed log(1/100) into -2 * log(10), which is a much simpler form. We then analyzed the answer choices and confirmed that C. -2 log 10 was the correct answer.

More importantly, we explored why this matters. The power rule of logarithms is a versatile tool with applications in various fields, from science and engineering to finance and computer science. By understanding and applying this rule, you can tackle complex problems and gain a deeper understanding of the world around you.

So, keep practicing, keep exploring, and keep those logarithmic skills sharp! You never know when they might come in handy. And remember, math isn't just about numbers and equations; it's about unlocking the power to understand and solve problems, making you a more informed and capable individual. Keep up the great work, guys!