Logarithms and antilogarithms are fundamental operations that facilitate the manipulation and analysis of exponential and multiplicative relationships through more manageable linear constructs.
However, most of the students often find it a difficult concept and have a hard time understanding the calculation of Logs. This happens because of the unclear concepts.
In this article, we aim to clarify the concepts of Log and Antilog and continue further to teach you to calculate them stepwise, in a hopefully more interesting manner.
What are Logarithms and Anti-logarithms?
Logarithm:
By definition:
“The logarithm (log) of a number is the exponent to which another fixed number, known as the base, must be raised to produce that number.”
In simpler terms, if you have an equation of the form by = x, then the logarithm of x with base b is y, written as y = logb(x).
Logarithms transform multiplicative relationships into additive and exponential relationships into linear. This makes them incredibly useful for simplifying complex calculations, especially before the era of calculators and computers.
Anti-Logarithm:
The antilogarithm is essentially the reverse process of taking a logarithm. If y = logb(x), then x is the antilogarithm of y to the base b, which is written as x=by. This operation is used to find the original number before it was transformed by the logarithm.
Log and Antilog through an interesting analogy:
Imagine you're a time traveler moving between different eras, where each era represents a scale of magnitude in the universe of numbers.
The Concept of Logarithms:
Think of logarithms as the way to calculate how far back in time (or forward) you need to travel to reach a certain era. The base of the logarithm is like your time machine’s settings whether it’s set to measure in centuries, millennia, or light-years, which drastically changes how you perceive distances between events.
Example:
Finding the logarithm of 100 with a base of 10(log10(100)) is like asking, "How many centuries do I need to go back in time to reach the year 1 from the year 100?" The answer is 2 centuries. In mathematical terms, you're raising the base (10) to the power of 2 to get 100, which means 10(log10(100)) = 2
The Concept of Antilogarithms:
Now, antilogarithms are like knowing you want to travel exactly 2 centuries into the past and trying to figure out which year you’ll end up in, starting from year 1. If your time machine’s setting (base) is centuries and you dial it to 2, you’re effectively asking, "Where does 2 centuries take me?"
Example:
The antilogarithm of 2 in base 10(102) tells you you’ll land in the year 100, starting from year 1. In this sense, taking the antilog of 2 with a base of 10 reverses the process, pinpointing the exact "era" (or number) you were curious about.
Basics for calculating Log and Antilog:
Before the widespread use of calculators (scientific and online antilog calculators), mathematicians and scientists used logarithm tables or slide rules for this purpose. However, to learn the manual process it is important to understand the base, log tables and key properties of the logs.
Understanding the Base:
● Common Logarithm: Has a base of 10. It's often written as
log(x) without specifying the base. This is widely used in engineering and common calculations.
● Natural Logarithm: Uses the base e (where e is an irrational constant approximately equal to 2.71828). Denoted as ln(x), it's fundamental in calculus, economics, and certain natural phenomena like growth processes.
Log table:
A logarithmic table is used to find the logarithm of a number without directly calculating it. The table lists the logarithms of numbers, usually in base 10. For numbers between 1 and 10, the logarithm table provides a direct lookup. For numbers outside this range, the properties of logarithms are used.
To find the logarithm of a number using a traditional log table:
Identify the Mantissa: The mantissa is the decimal part of the logarithm and is found by looking up the first few digits of the number in the table. The rest of the digits determine the precision.
Determine the Characteristic: The characteristic is the integer part of the logarithm and is determined by the number of digits to the left of the decimal in the original number.
● For numbers greater than 1, the characteristic is one less than the number of digits to the left of the decimal point.
● For numbers less than 1, the characteristic is negative (or in a form that reflects this for tables designed to avoid negative numbers) and depends on the number of leading zeros.
Logarithm Rules:
● Product Rule: The logarithm of a product of two numbers is the sum of the logarithms of the individual numbers.
logb(xy) = logb(x) + logb(y)
● Quotient Rule: The logarithm of a quotient of two numbers is the difference between the logarithm of the numerator and the logarithm of the denominator.
logb(x/y) = logb(x) - logb(y)
● Power Rule: The logarithm of a number raised to a power is simple power multiple with the logarithm of the number.
logb(xn) = n logb(x)
Log property:
● Change of Base Formula: The logarithm with one base can be converted to a logarithm with another base by using the change of base formula.
logb(x) = logk(x) / logk(b)
● Zero property: The logarithm of 1 to any base is 0 because any number raised to the power of 0 is 1.
logb(1) = 0
● Base Identity: The logarithm of the base itself is always 1.
logb(b) = 1
How to calculate Log and Antilog manually?
Now that your concepts are clear and you know the key factors involved in the calculation, let’s see the step-wise guide to calculate the Logs and Antilog.
Calculating Logs:
Step 1: Understand the Relationship.
Recall that if by = x, then logb(x) = y. This means you're looking for the power y to which the base b must be raised to get x.
Step 2: Use a Logarithm Table (for base 10 or e).
● Identify the Number: Break down your number into a significant digit(s) and a power of 10. For example, to find
log10(150), recognize it as 1.5 × 102.
Look Up the Significant Digit: Find the log value for the significant digit (1.5 in our example) in the table. Log tables typically list values for 1.00 to 9.99.
Adjust for the Power of 10: Add the power of 10 to the log value you found. For log10(150), you add 2 to the log value of 1.5 because 150=1.5×102.
Step 3: Interpolation (if necessary):
If your significant digit isn't exactly in the table, use linear interpolation between the closest values that are listed.
Example:
Determine the Logarithm of 2.
solution:
Assuming you have a log table that provides the mantissa for numbers from 1 to 10. To find log10(2):
Look up the mantissa for 2.00 in the table (you might find a value close to 0.3010).
Since 2 is a number greater than 1 but less than 10, its characteristic is 0 (because it has only one digit before the decimal).
Combine the characteristic and the mantissa:
log10(2) = 0.3010.
Example:
For a number like 200, which is 2 × 102, then find its logarithm value.
Solution:
● The mantissa is the same as for 2 (since 200 = 2 × 102), which is 0.3010.
● The characteristic is 2 (since there are three digits, the characteristic is 2).
● Thus, log10(200) = 2 + 0.3010 = 2.3010.
log10(200)= 2.3010.
Alternatively, you can use a log calculator by Allmath to save time and achieve more precise calculations.
Calculating Antilog:
Step 1: Understand the Relationship.
Given logb(x) = y, to find the antilogarithm means to solve for by=x.
Step 2: Use an Antilogarithm Table (if available for base 10 or e):
Identify the Logarithm: Split it into the integer part (characteristic) and the decimal part (mantissa).
Get original number: Multiply the number you found by 10{logarithm number} to get the original number.
Example:
Determine the Antilogarithm value of “0.3010” with base “10”.
Solution:
First, looking at the characteristic & mantissa part, then characteristic part zero and mantissa is “3010”.
Here, “0.3010” represents the logarithm of some number with base 10. Since we can write the antilogarithm,
Antilog(0.3010) = 100.3010 = 1.99986
Note: there, 1.99986 is approximately close to positive integer “2”.
Thus,
Antilog(0.3010)=2.
Example:
Determine the antilog of 3.2010.
Solution:
the antilog of 3.2010 can be found as:
The mantissa & characteristic part of “3.2010” is “2010” & “3” respectively.
Here, 3.2010 shows the log value, then the antilog can be written as:
Antilog(3.2010) = 103.2010 = 1588.54
This will be approximately nearly to the “1600”.
Thus,
Antilog(3.2010)=1600.
Alternatively, you use the Antilog calculator by Allmath, to make your calculation easy and faster than manual calculations or save your time.
Example Logarithmic Table
Here, we let's consider the logarithms of numbers from 1.00 to 1.10. The values are rounded to 4 decimal places.
Practical Applications
Logarithms Application: Useful in solving exponential and logarithmic equations, understanding scales that measure intensity (like the Richter scale for earthquakes), and in financial calculations involving compound interest or exponential growth.
Antilogarithms Application: Essential when we need to revert to the original scale from a logarithmic value. For example, after calculating logarithms for multiplication, using antilogarithms helps get back the product of the original numbers.