Logarithm Calculator: The Log Table Method

An interactive tool to understand how to calculate logarithms using log tables, breaking down the characteristic and mantissa.

Logarithm Result (log_b(x))

---

Scientific Notation

Characteristic

Mantissa

Results copied!

What is Calculating Logarithms Using Log Tables?

Before electronic calculators became common, mathematicians and scientists relied on printed log tables to perform complex multiplications and divisions. The process of using a log table involves breaking down a logarithm into two parts: the characteristic and the mantissa. This calculator simulates that historical method to help you understand how to calculate logarithms using log tables.

A logarithm answers the question: “what exponent do we need to raise a ‘base’ to, to get another number?”. For example, the logarithm of 100 to base 10 is 2, because 10² = 100. Log tables simplified this by providing pre-calculated values for the fractional part of the logarithm (the mantissa).

The Logarithm Formula and Explanation

The core principle when you want to how to calculate logarithms using log tables is to first express the number in scientific notation and then combine its parts. The general form of a logarithm is:

log_b(x) = y, which is equivalent to b^y = x

When using the log table method (typically for base 10), the logarithm is found by combining two components:

log₁₀(x) = Characteristic + Mantissa

If the base is not 10, the change of base formula is required: log_b(x) = log₁₀(x) / log₁₀(b).

Variables Table

Description of variables used in logarithm calculations.

Variable	Meaning	Unit	Typical Range
x	The input number.	Unitless (or depends on context)	Any positive real number
b	The base of the logarithm.	Unitless	Any positive real number ≠ 1
Characteristic	The integer part of the logarithm, representing the order of magnitude.	Integer	-∞ to +∞
Mantissa	The fractional part of the logarithm, derived from the significant digits of the number.	Decimal	0 to 1

A plot of the logarithmic function, highlighting the calculated point.

Practical Examples

Example 1: Common Logarithm (Base 10)

Let’s find the logarithm of 316.2 using the log table method.

• Inputs: Number (x) = 316.2, Base (b) = 10
• Step 1: Scientific Notation: 316.2 is written as 3.162 x 10².
• Step 2: Find Characteristic: The exponent of 10 is 2. So, the characteristic is 2.
• Step 3: Find Mantissa: We would look up log₁₀(3.162) in a log table. Our calculator does this mathematically, giving approximately 0.5.
• Result: The final logarithm is Characteristic + Mantissa = 2 + 0.5 = 2.5. (Note: 10^2.5 ≈ 316.2)

Example 2: Change of Base

Let’s calculate log₂(64) using the principles of the change of base formula.

• Inputs: Number (x) = 64, Base (b) = 2
• Step 1: Use Change of Base Formula: log₂(64) = log₁₀(64) / log₁₀(2).
• Step 2: Calculate Numerator (log₁₀(64)):
  • Scientific Notation: 6.4 x 10¹.
  • Characteristic = 1. Mantissa = log₁₀(6.4) ≈ 0.806.
  • log₁₀(64) ≈ 1.806.
• Step 3: Calculate Denominator (log₁₀(2)):
  • Scientific Notation: 2.0 x 10⁰.
  • Characteristic = 0. Mantissa = log₁₀(2.0) ≈ 0.301.
  • log₁₀(2) ≈ 0.301.
• Result: Divide the two results: 1.806 / 0.301 ≈ 6.

How to Use This Logarithm Calculator

This calculator makes it easy to understand the components of a logarithm. Here’s a step-by-step guide:

1. Enter the Number (x): Input the positive number you want to find the logarithm for in the first field.
2. Enter the Base (b): Input the base of the logarithm. For a common logarithm calculator, use 10. For a natural logarithm, use ‘e’ (approx. 2.71828).
3. Review the Results: The calculator automatically updates. The main result is the final logarithm value.
4. Understand the Intermediate Values: The calculator shows the ‘Scientific Notation’ of your number (for base 10), the ‘Characteristic’ (the integer part), and the ‘Mantissa’ (the fractional part). This breakdown is the essence of how log tables work. For more powerful calculations, you can use a full scientific calculator.

Sample Log Table

Below is a simplified table showing how one might find the mantissa for numbers between 1.0 and 1.4. In a real log table, you would find the row for ‘1.2’ and the column for ‘3’ to find the mantissa for 1.23.

A small sample of a base-10 logarithm table. The values shown are the mantissas.

N	0	1	2	3	4
1.0	.0000	.0043	.0086	.0128	.0170
1.1	.0414	.0453	.0492	.0531	.0569
1.2	.0792	.0828	.0864	.0899	.0934
1.3	.1139	.1173	.1206	.1239	.1271
1.4	.1461	.1492	.1523	.1553	.1584

Key Factors That Affect Logarithms

• Magnitude of the Number: Larger numbers have larger logarithms.
• The Base: A larger base leads to a smaller logarithm for numbers greater than 1. For instance, log₁₀(1000) is 3, but log₁₀₀(1000) is 1.5.
• Numbers Between 0 and 1: Numbers between 0 and 1 always have negative logarithms (for bases greater than 1).
• Scientific Notation Exponent: The exponent in the scientific notation of a number directly determines the characteristic of its base-10 logarithm.
• Significant Digits: The significant digits of a number (e.g., the ‘3162’ in 316.2) determine the mantissa of its logarithm. Learn more about understanding exponents.
• Logarithm of 1: The logarithm of 1 is always 0, regardless of the base, because any base raised to the power of 0 is 1.

FAQ

What is the difference between characteristic and mantissa?

The characteristic is the integer part of a common logarithm, indicating the number’s magnitude (power of 10). The mantissa is the non-negative decimal part, determined by the sequence of digits in the number. This is a core concept for the logarithm characteristic mantissa method.

Why do we use base 10 for log tables?

Base 10, or the “common logarithm”, was used because our number system is base-10. This makes it easy to determine the characteristic by simply counting the digits or finding the power of 10 in scientific notation.

How do I find the log of a number with a different base, like base 2?

You must use the change of base formula: log_b(x) = log_c(x) / log_c(b). You can convert your logarithm to any new base ‘c’, but typically you convert to base 10 or base ‘e’ (natural log) to use standard tables or calculators.

Can you calculate the logarithm of a negative number?

No, the logarithm is not defined for negative numbers or zero in the set of real numbers. The input must be a positive number.

What is an antilog?

An antilog (or antilogarithm) is the inverse of a logarithm. It’s the number you get when you raise the base to the power of the logarithm. For example, the antilog of 2 (base 10) is 10², which is 100. Check out our antilog calculator for more.

What is the difference between log and ln?

“log” usually implies base 10 (common logarithm), while “ln” refers to the natural logarithm, which has a base of ‘e’ (approximately 2.718).

Are log tables still used today?

No, log tables have been completely replaced by scientific calculators and computers. However, understanding how they work provides great insight into the properties of logarithms and is a valuable topic in mathematics education.

How would you find log(0.05)?

First, write it in scientific notation: 5 x 10^-2. The characteristic is -2. The mantissa is log₁₀(5) ≈ 0.699. The result is -2 + 0.699 = -1.301.

Related Tools and Internal Resources

Explore these other tools and guides to expand your knowledge of mathematics and conversions:

• Antilog Calculator: The inverse operation of this calculator. Find the original number from its logarithm.
• What is a Logarithm?: A detailed guide explaining the concept from the ground up.
• Scientific Calculator: For more complex calculations involving a variety of mathematical functions.
• Comprehensive Math Formulas: A reference for various mathematical formulas, including logarithmic properties.
• Understanding Exponents: A foundational guide to exponents, which are intrinsically linked to logarithms.
• Base Converter: Convert numbers between different numeral systems (like binary, decimal, hex).