Finding The Common Logarithm Of 75 Using A Calculator

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In the realm of mathematics, logarithms play a crucial role in simplifying complex calculations and solving exponential equations. Among the various types of logarithms, common logarithms, also known as base-10 logarithms, hold a significant position due to their direct connection with the decimal system. In this comprehensive guide, we will delve into the concept of common logarithms, explore their properties, and demonstrate how to utilize calculators to determine the common logarithm of a given number, specifically 75.

Unveiling the Essence of Common Logarithms

At its core, a logarithm is the inverse operation of exponentiation. In simpler terms, the logarithm of a number to a given base is the exponent to which the base must be raised to produce that number. Common logarithms, denoted as log₁₀(x) or simply log(x), employ 10 as their base. This means that the common logarithm of a number x represents the power to which 10 must be raised to obtain x.

For instance, the common logarithm of 100 is 2 because 10 raised to the power of 2 equals 100 (10² = 100). Similarly, the common logarithm of 1000 is 3 because 10³ = 1000. Understanding this fundamental relationship between exponentiation and logarithms is essential for grasping the concept of common logarithms.

The significance of common logarithms stems from their alignment with the decimal system, which is the foundation of our everyday numerical system. This alignment makes common logarithms particularly convenient for various applications, including scientific calculations, engineering problems, and financial analysis. Moreover, common logarithms played a pivotal role in the pre-calculator era, as they facilitated complex calculations through the use of logarithm tables.

Properties of Common Logarithms

Common logarithms possess several key properties that make them a powerful tool in mathematical manipulations. These properties include:

  1. Product Rule: The logarithm of the product of two numbers is equal to the sum of their individual logarithms. Mathematically, this is expressed as:

    log(xy) = log(x) + log(y)

  2. Quotient Rule: The logarithm of the quotient of two numbers is equal to the difference of their individual logarithms. The mathematical representation is:

    log(x/y) = log(x) - log(y)

  3. Power Rule: The logarithm of a number raised to a power is equal to the product of the power and the logarithm of the number. This can be written as:

    log(x^n) = n * log(x)

These properties are invaluable in simplifying logarithmic expressions and solving logarithmic equations. By applying these rules, complex calculations can be broken down into simpler steps, making them more manageable.

Harnessing Calculators to Determine Common Logarithms

In the modern era, calculators have become indispensable tools for mathematical calculations, including finding common logarithms. Most scientific calculators come equipped with a dedicated logarithm function, usually denoted as "log" or "log₁₀". This function directly calculates the common logarithm of a given number.

To find the common logarithm of 75 using a calculator, follow these straightforward steps:

  1. Locate the logarithm function: Identify the "log" or "log₁₀" key on your calculator. This key typically resides among the other mathematical functions.

  2. Enter the number: Input the number 75 into the calculator's display.

  3. Press the logarithm key: Press the "log" or "log₁₀" key. The calculator will perform the calculation and display the result.

The calculator will output the common logarithm of 75, which is approximately 1.87506. This value signifies that 10 raised to the power of 1.87506 is approximately equal to 75.

Illustrative Examples

To further solidify your understanding, let's explore a few more examples of using calculators to determine common logarithms:

  • Find the common logarithm of 250:

    Input 250 into the calculator and press the "log" key. The result will be approximately 2.39794.

  • Determine the common logarithm of 0.5:

    Enter 0.5 into the calculator and press the "log" key. The result will be approximately -0.30103.

  • Calculate the common logarithm of 10000:

    Input 10000 into the calculator and press the "log" key. The result will be 4, as expected, since 10⁴ = 10000.

These examples showcase the ease and efficiency of using calculators to find common logarithms. With a few simple steps, you can determine the common logarithm of any positive number.

Applications of Common Logarithms

Common logarithms find widespread applications in various fields, including:

  • Science: Common logarithms are used in chemistry to express pH values, which measure the acidity or alkalinity of a solution. They are also employed in physics to represent the magnitude of earthquakes on the Richter scale.

  • Engineering: Common logarithms are utilized in electrical engineering to calculate decibel values, which measure the intensity of sound or electrical signals. They also find applications in structural engineering for analyzing stress and strain in materials.

  • Finance: Common logarithms are used in financial analysis to calculate compound interest and determine the time it takes for an investment to double in value.

  • Mathematics: Common logarithms serve as a fundamental tool in solving exponential equations and simplifying complex mathematical expressions.

The versatility of common logarithms makes them an indispensable tool for scientists, engineers, financial analysts, and mathematicians alike.

Conclusion

In this comprehensive exploration, we have unraveled the concept of common logarithms, delved into their properties, and demonstrated how to utilize calculators to determine the common logarithm of a given number, specifically 75. We have also highlighted the diverse applications of common logarithms in various fields, underscoring their significance in scientific, engineering, financial, and mathematical contexts.

By mastering the concept of common logarithms and their practical applications, you will gain a valuable tool for simplifying complex calculations and solving exponential equations. Whether you are a student, a professional, or simply an inquisitive mind, understanding common logarithms will undoubtedly enhance your mathematical prowess and problem-solving abilities.

Practice Exercises

To further solidify your understanding, try solving these practice exercises:

  1. Find the common logarithm of 150.
  2. Determine the common logarithm of 0.25.
  3. Calculate the common logarithm of 5000.
  4. Solve for x: log(x) = 2.5.
  5. Solve for x: log(x) + log(2) = log(10).

By working through these exercises, you will reinforce your knowledge and gain confidence in using calculators to find common logarithms.

For further exploration of common logarithms, consult these resources:

  • Khan Academy: Logarithms
  • Mathway: Logarithm Calculator
  • WolframAlpha: Common Logarithm

These resources offer additional explanations, examples, and tools to deepen your understanding of common logarithms.

By leveraging these resources and diligently practicing the concepts discussed in this guide, you will unlock the full potential of common logarithms and enhance your mathematical capabilities.