Finding the logarithm of a decimal number involves separating it into two distinct parts: the characteristic (the integer part) and the mantissa (the positive fractional part). This method simplifies calculating logarithms for numbers both greater and smaller than one.
Understanding Logarithms of Decimal Numbers
A logarithm answers the question: "To what power must a base be raised to get a certain number?" For decimal numbers, especially when using common logarithms (base 10), the process involves analyzing the number's magnitude to determine its characteristic, and then finding its mantissa based on its significant digits.
Components of a Logarithm
Every logarithm can be expressed as the sum of its characteristic and mantissa.
The Characteristic
The characteristic is the integer part of the logarithm and indicates the order of magnitude of the original number. Its value depends on whether the decimal number is greater than or less than 1.
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For Numbers Greater Than or Equal to 1 (N ≥ 1):
The characteristic is determined by the number of digits before the decimal point. You subtract one from this count.- Rule: Characteristic = (Number of digits before the decimal point) - 1
- Example: For the number 342.2, there are 3 digits (3, 4, 2) before the decimal point.
Therefore, the characteristic is 3 - 1 = 2.
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For Numbers Between 0 and 1 (0 < N < 1):
The characteristic is negative and is determined by the number of zeros immediately following the decimal point. You count these zeros, add one, and then make the result negative.- Rule: Characteristic = - (Number of zeros immediately after the decimal point + 1)
- Example: For the number 0.3422, there are 0 zeros immediately after the decimal point.
Therefore, the characteristic is - (0 + 1) = -1. - Example: For 0.005, there are 2 zeros after the decimal point. The characteristic would be -(2+1) = -3.
The Mantissa
The mantissa is the positive fractional part of the logarithm. It is determined by the sequence of significant digits of the number, regardless of the decimal point's position. The mantissa is always positive.
Historically, mantissas were found using logarithm tables. Today, scientific calculators or online tools compute them directly. For example, the mantissa for the digits 3422 (whether from 342.2 or 0.3422) is approximately 0.5343.
Calculating the Logarithm: Putting It Together
Once you have both the characteristic and the mantissa, you simply add them together to find the complete logarithm of the decimal number:
Logarithm = Characteristic + Mantissa
Examples of Calculating Decimal Logs
Let's illustrate with the provided examples:
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Example 1: Finding log(342.2)
- Determine the Characteristic: The number 342.2 has 3 digits before the decimal point.
Characteristic = 3 - 1 = 2. - Identify the Mantissa: For the digits 3422, the mantissa is 0.5343.
- Combine: Add the characteristic and mantissa.
log(342.2) = 2 + 0.5343 = 2.5343.
- Determine the Characteristic: The number 342.2 has 3 digits before the decimal point.
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Example 2: Finding log(0.3422)
- Determine the Characteristic: The number 0.3422 has 0 zeros immediately after the decimal point.
Characteristic = - (0 + 1) = -1. - Identify the Mantissa: For the digits 3422, the mantissa is 0.5343.
- Combine: Add the characteristic and mantissa. When the characteristic is negative, you perform the subtraction.
log(0.3422) = -1 + 0.5343 = -0.4657.
- Determine the Characteristic: The number 0.3422 has 0 zeros immediately after the decimal point.
Quick Reference Table for Characteristics
| Number (N) | Rule for Characteristic | Example Number | Characteristic |
|---|---|---|---|
| N ≥ 1 | (Number of digits before decimal) - 1 | 54.7 | 1 |
| 9 | 0 | ||
| 1,234.5 | 3 | ||
| 0 < N < 1 | - (Number of zeros immediately after decimal + 1) | 0.82 | -1 |
| 0.056 | -2 | ||
| 0.0001 | -4 |
Practical Tools for Logarithms
While understanding the characteristic and mantissa is fundamental, in modern practice, scientific calculators and online logarithm tools readily provide the complete logarithm for any decimal number. These tools automatically perform the separation and combination of these parts.
- To learn more about the basics of logarithms, you can explore resources like Khan Academy's Introduction to Logarithms.
