Calculating Log10 1.23 A Step By Step Guide

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In the realm of mathematics, logarithms play a pivotal role, especially in simplifying complex calculations and understanding exponential relationships. This article delves into the concept of logarithms, specifically focusing on how to calculate the value of log₁₀ 1.23 given that log₁₀ 123 = 2.09. We will explore the properties of logarithms, their applications, and provide a step-by-step guide to solving this particular problem. Whether you're a student grappling with logarithmic equations or simply a math enthusiast, this article aims to provide a comprehensive understanding of the subject.

What are Logarithms?

To begin, it’s crucial to understand what logarithms are and why they are essential in mathematics. A logarithm is essentially the inverse operation to exponentiation. In simpler terms, if we have an exponential equation like b^y = x, the logarithmic form of this equation is log_b(x) = y. Here, b is the base of the logarithm, x is the argument, and y is the exponent.

Logarithms help in simplifying calculations involving very large or very small numbers. They are used extensively in various fields such as physics, engineering, computer science, and economics. For instance, in physics, logarithms are used to measure the intensity of earthquakes (Richter scale) and the loudness of sound (decibels). In computer science, they are used in the analysis of algorithms.

Common Logarithms (Base 10)

Among the different bases for logarithms, the most commonly used is base 10, denoted as log₁₀ or simply log. When we write log(x) without specifying the base, it is generally understood to be base 10. Base 10 logarithms are particularly useful because our number system is also base 10, making them convenient for many everyday calculations.

Properties of Logarithms

Understanding the properties of logarithms is key to solving logarithmic problems. Here are some fundamental properties:

  1. Product Rule: log_b(mn) = log_b(m) + log_b(n)
  2. Quotient Rule: log_b(m/n) = log_b(m) - log_b(n)
  3. Power Rule: log_b(m^p) = p * log_b(m)
  4. Change of Base Rule: log_b(a) = log_c(a) / log_c(b)

These properties allow us to manipulate logarithmic expressions and simplify complex equations. They are the tools we will use to solve the problem at hand.

Problem Statement: Calculating log₁₀ 1.23 Given log₁₀ 123 = 2.09

Now, let’s revisit the problem. We are given that log₁₀ 123 = 2.09, and we need to find the value of log₁₀ 1.23. This might seem daunting at first, but by using the properties of logarithms, we can break it down into manageable steps.

The key to solving this problem lies in recognizing the relationship between 123 and 1.23. We can express 1.23 as 123 divided by 100. This insight allows us to use the quotient rule of logarithms.

Step-by-Step Solution

Here’s how we can solve the problem step by step:

Step 1: Express 1.23 as a Fraction

As mentioned earlier, we can express 1.23 as a fraction of 123. Specifically, 1.23 = 123 / 100. This is a crucial step because it allows us to use the quotient rule of logarithms.

Step 2: Apply the Quotient Rule

The quotient rule states that log_b(m/n) = log_b(m) - log_b(n). Applying this rule to our problem, we get:

log₁₀ 1.23 = log₁₀ (123 / 100) = log₁₀ 123 - log₁₀ 100

Step 3: Substitute the Given Value

We are given that log₁₀ 123 = 2.09. So, we substitute this value into our equation:

log₁₀ 1.23 = 2.09 - log₁₀ 100

Step 4: Evaluate log₁₀ 100

Now, we need to find the value of log₁₀ 100. Remember that log₁₀ 100 is asking the question: “To what power must we raise 10 to get 100?” The answer is 2, since 10ÂČ = 100. Therefore, log₁₀ 100 = 2.

Step 5: Final Calculation

Substitute log₁₀ 100 = 2 into our equation:

log₁₀ 1.23 = 2.09 - 2

log₁₀ 1.23 = 0.09

So, the value of log₁₀ 1.23 is 0.09.

Why This Matters: Practical Applications of Logarithms

Understanding how to calculate logarithms is not just an academic exercise; it has practical applications in various fields. Let’s explore some of these applications to appreciate the real-world significance of logarithms.

1. Earthquake Intensity (Richter Scale)

The Richter scale, used to measure the magnitude of earthquakes, is a logarithmic scale. The magnitude M of an earthquake is given by:

M = log₁₀ (I / S)

where I is the amplitude of the seismic waves measured by a seismograph, and S is the amplitude of a standard earthquake. The logarithmic nature of the scale means that an increase of one unit on the Richter scale corresponds to a tenfold increase in the amplitude of the seismic waves. This makes logarithms indispensable for quantifying and comparing earthquake intensities.

2. Sound Intensity (Decibels)

The loudness of sound is measured in decibels (dB), which also uses a logarithmic scale. The sound level L in decibels is given by:

L = 10 * log₁₀ (I / I₀)

where I is the intensity of the sound, and I₀ is the reference intensity (the threshold of hearing). Similar to the Richter scale, the logarithmic scale allows us to represent a wide range of sound intensities in a manageable way. A small increase in decibels corresponds to a significant increase in sound intensity, which is why logarithmic scales are used.

3. Chemistry (pH Scale)

The pH scale, used to measure the acidity or alkalinity of a solution, is another example of a logarithmic scale. The pH is defined as:

pH = -log₁₀ [H+]

where [H+] is the concentration of hydrogen ions in moles per liter. A pH of 7 is neutral, values below 7 are acidic, and values above 7 are alkaline. The logarithmic scale makes it easy to express a wide range of hydrogen ion concentrations.

4. Computer Science (Algorithm Analysis)

In computer science, logarithms are used in the analysis of algorithms. For example, the time complexity of binary search is O(log n), where n is the number of elements. This means that the time taken to search for an element doubles as the number of elements doubles, making it a very efficient search algorithm for large datasets.

5. Finance (Compound Interest)

Logarithms are used in financial calculations, particularly in compound interest problems. The formula for compound interest is:

A = P (1 + r/n)^(nt)

where A is the final amount, P is the principal amount, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the number of years. Logarithms can be used to solve for t or r when the other variables are known.

Common Mistakes to Avoid

When working with logarithms, it’s easy to make mistakes if you’re not careful. Here are some common pitfalls to avoid:

  1. Incorrectly Applying Logarithmic Properties: Make sure you understand and correctly apply the product, quotient, and power rules. A common mistake is to assume that log(a + b) = log(a) + log(b), which is incorrect. The correct rule is log(ab) = log(a) + log(b).
  2. Forgetting the Base: Always remember the base of the logarithm. If the base is not explicitly written, it is generally assumed to be 10. However, in some contexts, it might be the natural logarithm (base e).
  3. Confusing Logarithms with Exponents: Logarithms and exponents are inverse operations, but they are not the same. Make sure you understand the relationship between them and use the correct notation.
  4. Dividing Inside the Logarithm: Always double-check if you're dividing inside or outside the logarithm. For instance, log(x/y) is not the same as log(x) / log(y).

By being aware of these common mistakes, you can improve your accuracy when solving logarithmic problems.

Conclusion

In summary, understanding logarithms and their properties is crucial for solving mathematical problems and for various applications in science and engineering. We have shown how to calculate log₁₀ 1.23 given that log₁₀ 123 = 2.09 by using the quotient rule of logarithms. This problem illustrates the power and versatility of logarithms in simplifying calculations.

Logarithms are more than just a mathematical concept; they are a tool that helps us understand and quantify phenomena in the world around us. From measuring earthquake intensities to analyzing algorithms, logarithms play a vital role in various fields. By mastering the properties and applications of logarithms, you can enhance your problem-solving skills and gain a deeper appreciation for the beauty and utility of mathematics.

Remember, the key to success with logarithms is practice. Work through various problems, apply the properties, and you’ll become proficient in no time. Whether you’re a student, engineer, or simply a math enthusiast, logarithms are a valuable tool in your arsenal.

To reinforce your understanding of logarithms, here are a few additional practice problems:

  1. Given log₁₀ 2 ≈ 0.3010, find log₁₀ 200.
  2. If log₂ x = 5, what is the value of x?
  3. Solve for x: log₃ (2x + 1) = 2.
  4. Simplify: log₅ 125 - log₅ 5.
  5. If log₁₀ 5 = 0.6990, find log₁₀ 25.

Work through these problems, and you’ll become even more confident in your ability to work with logarithms. Happy calculating!

This article has provided a detailed exploration of logarithms, their properties, and their applications. By understanding the fundamental concepts and practicing regularly, you can master logarithms and use them effectively in various contexts. Keep exploring, keep learning, and you’ll discover the fascinating world of mathematics and its endless possibilities.