Logarithmic Arithmetic

Required Prerequisites- Exponential Arithmetic
Today, the logarithm is taught mostly in context of subjects like compound interest, decay rates, sound pressures, and earthquake intensities. Though useful, these subjects do not give the logarithm the credit it is due. From 1624 (when Henry Briggs published Arithmetica Logarithmica) until 1972 (when Hewlett-Packard release the first scientific pocket calculator), the logarithm was the primary means by which most of the world performed arithmetic. Logarithms were used to design airplanes, create the atomic bomb, send men to the moon, and so much more. If not for the discovery of logarithms, we would not be enjoying any of the modern technological marvels we have today.
Multiplication and Division are Hard!
Subtraction and addition by hand are relatively easy, but division and multiplication are hard. Even simple calculations like


Figure 1: Left- Long division demonstrating
If only multiplication and division could be simpler... as we will see shortly, with the power of logarithms, they can be!
The Logarithm Definition
To derive the logarithm, we will be using exponential arithmetic extensively, I recommend reviewing it.
During exponentiation, we multiply a base number
In a very simple example,
Consider the relationship
What is the value for
Often, this isn't such an easy problem. If
This problem of finding the power
Consider the area of a square. The area
This equation has an input and output, each on separate sides of the equality sign:
The input of the equation is on the left. The output on the right. This equation is neat and concise, but in the problem of solving for the power in an exponential, it is not so easy to separate the inputs
The base-
For example, consider the exponential
If we take the base-
From the fundamental definition,
We now have an equation with the inputs
Taking the base-
This process, left to right, is the logarithm transformation.
Consider the situation where we take the base number
Since
This equation, obtained by exponentiating
We now have the fundamental definition for the logarithm transformation and how to reverse it:
When
The solution for this equation is
Our First Logarithm Table
Suppose that for the base

To convert the exponential table to a logarithm table, we use the logarithm transformation equation to relabel the table's columns so that

Since the contents of the exponential and logarithmic tables are identical, proper labeling of the columns permit the same table to be used for both. The table is most useful for us with

Each row in the
Using the base-
Easy Multiplication using Logarithms
Using logarithms, we can multiply two exponential numbers
Taking the base-
If we multiply together
Using the logarithm definition, the base-
Since
This fundamental logarithm property is called the product rule.
A practical use of this property can be demonstrated with an example. We desire to calculate
Using the product rule property, we find:
Now, the base
Using the base-
The problem of performing multiplication has been reduced to using a table and addition!
Easy Division using Logarithms
Using logarithms, we can divide two exponential numbers
Taking the base-
When dividing
Using the logarithm definition, the base-
Since
This fundamental logarithm property is called the quotient rule.
A practical use of this property can be demonstrated with an example. We desire to calculate
Using the quotient rule property, we find:
Now, the base
Using the base-
The problem of performing division has been reduced to using a table and subtraction!
Easy Exponentiation and Roots Using Logarithms
Recall the logarithm transformation:
Raising both sides of
Using the power of power rule, we obtain:
Taking the base-
Multiplying both sides of
Since
This fundamental logarithm property is called the power rule.
A practical use of this property can be demonstrated with an example. We desire to calculate
From the base-
Therefore:
Exponentiating
Using the base-
Incredibly, the problem of finding a cubed root has been reduced to using a table and division!
Logarithms of Negative Numbers
Suppose we have a base number
Unfortunately, this equation has no solution. This means that the logarithm
Though the logarithm of a negative number is undefined, the inverse logarithm remains valid when applied to the logarithm of a negative number:
This fundamental logarithm property is called the round-trip rule.
The round-trip rule states that as long as we don't try to calculate the value of the logarithm of a negative number, negative numbers can still be taken "round-trip" through the logarithm transformation:
As an example, we can take the number
Any negative number
Taking the base-
Exponentiating the base
Exponentiation rules permit the exponent to be split:
Simplifying, we obtain:
As has been shown, both
Taking the logarithm of both sides:
Exponentiating the base
Simplifying, we obtain:
When performing multiplication or division, negative signs are preserved and can be propagated without needing to track the negative sign through the round-trip process.
We can also use logarithms to calculate the value of exponentiated negative numbers, though we've not yet arrived at a point where doing so is very useful. First, the negative number should be separated into positive and base-
After separation, the positive root can be calculated as normal using logarithm rules and tables,
A Clarification on the Existence of Logarithms for Negative Numbers
This is an advanced note. Unless you are a devout mathematics enthusiast, you can likely ignore this clarification.
To say that the logarithm of a negative number is undefined is not entirely correct. If the result of the logarithm is required to be a real number, then the logarithm of a negative number is indeed undefined. However, if we permit the result of a logarithm to be a complex number, then the logarithm of a negative number does exist.
Though the logarithm of a negative number technically does exist, the round-trip rule means it's not typically needed for conventional arithmetic. Because calculating logarithms of negative numbers is a more complex topic, it will be left for a future discussion.
The Base-10 Logarithm Table
Earlier, we created and used a base-3 logarithm table for a number of examples, but there isn't really anything special about the base-3 logarithm. As we will see shortly, the base-10 logarithm actually has a property that makes it excellent for arithmetic. Below is a simple base-

Consider we want to multiply two numbers,
While calculating the base-



Figure 2: Example pages from Henry Briggs' Arithmetica Logarithmica, 1624
The work was grueling, but the result was worth it. Briggs' initiated an acceleration in mathematics that today has gifted us with technology beyond his wildest imaginations.
If you'd like to download a PDF version of Arithmetic Logarithmica, the link is available below, source scans courtesy of https://archive.org/.
Scientific Notation
The aforementioned beneficial property of the base-
Below are some examples of decimal numbers expressed in integer scientific notation:
By converting a decimal number to integer scientific notation format, a logarithm table only needs to contain the logarithm value for integers.
Example of Multiplication with Base-10 Logarithm Table
We want to multiply two numbers,
Multiplying them together in scientific notation, we obtain:
Using the logarithm product rule:



Figure 3: Pages from Arithmetica Logarithmica for calculating
Using Briggs' logarithm table (figure 3):
Raising
When calculating exponentials, Briggs' table provides the most digits of precision for exponentials between
Using the base-
Therefore:
Note that the actual full and correct answer is
Even with all of our rounding, our calculation (which we calculated using only a large table and a single addition) had an error of
Example of Division with Base-10 Logarithm Table
Now consider the case where we desire to divide
Taking their quotient, we obtain:
Using the logarithm quotient rule:



Figure 4: Pages from Arithmetica Logarithmica for calculating
Using Briggs' logarithm table (figure 4):
Raising
When calculating exponentials, Briggs' table provides the most digits of precision for exponentials between
Using the base-
Therefore:
Note that the actual full and correct answer is
Even with all of our rounding, our calculation (which we calculated using only a large table and a single subtraction) has an error of