Understanding The Value Of Log 3

by Olex Johnson 33 views

Hello there! Let's dive into the fascinating world of logarithms and figure out the value of log 3. I'll provide you with a clear, detailed, and correct explanation to help you understand this concept thoroughly.

Correct Answer

The value of log 3 (using base 10) is approximately 0.4771.

Detailed Explanation

Let's break down what log 3 means and how we arrive at this value. Logarithms can seem a bit tricky at first, but once you understand the basics, they become quite manageable. The term log 3 refers to the base-10 logarithm of 3. The base is generally omitted, and it is assumed to be 10 unless otherwise specified (e.g., ln, which implies a base of e).

Key Concepts

  1. Logarithm: A logarithm answers the question: "To what power must we raise the base to get a certain number?"
  2. Base: The base of a logarithm is the number that is being raised to a power. If no base is explicitly written, it is assumed to be 10.
  3. Argument: The argument is the number we are taking the logarithm of. In log 3, the argument is 3.
  4. Logarithmic Form: The general form is logb(x) = y, where b is the base, x is the argument, and y is the exponent.
  5. Exponential Form: The equivalent exponential form is by = x.

Let's look at a simple example to illustrate these concepts. Consider log2(8) = 3. This means: "To what power must we raise 2 to get 8?" The answer is 3, because 23 = 8.

Now, let's apply these concepts to log 3. This is equivalent to asking: "To what power must we raise 10 to get 3?"

Since we can't easily calculate this in our heads like log2(8), we usually rely on a calculator or a logarithm table. Here's how you can think about it:

  1. Understanding the Range: We know that:

    • 100 = 1
    • 101 = 10
    • Since 3 lies between 1 and 10, the value of log 3 will be between 0 and 1.

  2. Using a Calculator:

    • Most scientific calculators have a log button. To find log 3, simply enter 3 and press the log button.
    • The result is approximately 0.4771.

  3. Interpreting the Result:

    • This means that 100.4771 ≈ 3. Or, when you raise 10 to the power of 0.4771, you get a number close to 3. It is an approximation because logarithms often involve irrational numbers.

Why Logarithms are Important

Logarithms might seem abstract, but they are incredibly useful in various fields:

  1. Science and Engineering: Logarithms are used to measure the intensity of sound (decibels), the brightness of stars (magnitudes), and the acidity of solutions (pH). For example, a pH of 7 represents neutral acidity. Values less than 7 are acidic, and values greater than 7 are basic. The scale is logarithmic, so a change of 1 pH unit represents a tenfold change in acidity.
  2. Computer Science: Logarithms are used in analyzing the efficiency of algorithms. For example, in computer science, the time complexity of certain algorithms (like binary search) is expressed using logarithms.
  3. Finance: Logarithms are used in calculations involving compound interest and in understanding the rate of return on investments. They help in modeling exponential growth.
  4. Earthquake Measurement: The Richter scale, used to measure the magnitude of earthquakes, is logarithmic. Each whole number increase on the Richter scale represents a tenfold increase in the measured amplitude of the seismic waves.

Example: Breaking Down Logarithms

Let's consider another example to solidify understanding. What is log 100?

  • This is asking: "To what power must we raise 10 to get 100?"
  • Since 102 = 100, log 100 = 2.

Another example: log 1000?

  • "To what power must we raise 10 to get 1000?"
  • Since 103 = 1000, log 1000 = 3.

Common Logarithms vs. Natural Logarithms

It's also important to understand the difference between common logarithms and natural logarithms.

  1. Common Logarithms: These use base 10 and are typically written as log x.
  2. Natural Logarithms: These use the base e (Euler's number, approximately 2.71828) and are written as ln x.
    • The natural logarithm is used extensively in calculus and many scientific applications.
    • The relationship between log and ln is: ln x = (log x) / (log e).

Laws of Logarithms

Understanding the laws of logarithms helps in simplifying and solving logarithmic equations:

  1. Product Rule: logb(xy) = logb(x) + logb(y)
    • Example: log2(8 * 4) = log2(8) + log2(4)
  2. Quotient Rule: logb(x/y) = logb(x) - logb(y)
    • Example: log2(8/4) = log2(8) - log2(4)
  3. Power Rule: logb(xn) = n logb(x)
    • Example: log2(42) = 2 * log2(4)
  4. Change of Base Formula: logb(x) = loga(x) / loga(b)
    • This allows us to convert logarithms from one base to another.

Applying these rules makes solving complex logarithmic equations much easier.

Practice Problems

Try these practice problems to reinforce your understanding:

  1. What is log 10000?
  2. What is ln e?
  3. If log x = 2, what is x?

Answers:

  1. log 10000 = 4 (since 104 = 10000)
  2. ln e = 1 (since e1 = e)
  3. x = 100 (since 102 = 100)

Key Takeaways

  • log 3 (base 10) is approximately 0.4771.
  • A logarithm answers the question: "To what power must we raise the base to get a certain number?"
  • Logarithms are essential in science, engineering, computer science, and finance.
  • Common logarithms use base 10 (log x), while natural logarithms use base e (ln x).
  • Understanding the rules of logarithms simplifies calculations and equations.

I hope this detailed explanation has been helpful! Keep practicing, and you'll become a logarithm expert in no time! If you have more questions, feel free to ask.