The logarithm of a number n refers to the number of times another number called the base, or b must be repeatedly multiplied to produce n. In other words, what the base b must be raised to get the number n is called n’s logarithm. If a number b must be raised to the power of y to produce the number x or if b^y = x, then its logarithmic form would be log_b x = y. Technically, logarithms undo what exponentiation does.

What are Logarithms?

Logarithms are, in fact, the inverse function of exponentiation, just as the mathematical operations of multiplication and division are the inverse functions of each other.

When we multiply 10 by 10, or when we raise 10 to the power of 2, we get 100. While this is stated by the exponential expression 10² = 100 (read as ten raised to 2 is equal to hundred), another way of stating this is the logarithmic expression, log₁₀ 100 = 2 (read as the log of hundred to the base ten is equal to two).

Basically,
Exponential expression: 10² = 100 (10 raised to 2 is 100)
Logarithmic expression: log₁₀ 100 = 2 (log of 100 to the base 10 is 2)

Logarithm Example:

Q.1)
• Convert the following in the logarithmic form:
a) 100^½ = 10
b) 32^0.2 = 2
• Convert the following in the exponential form:
a) log₃ 1/81 = -4
b) log₂₇ 9 = 2/3
Solution:
• a) log₁₀₀ 10 = 1/2
b) log₃₂ 2 = 0.2
• a) 3^-4 = 81
b) 27^2/3 = 9

The argument can be written encased in round brackets or without them. For instance, log₂ 4 can also be written as log₂(4).
When the logarithms of arguments are written without a base, the base is usually assumed to be 10, by default, unless the base is specifically stated or mentioned. For instance, log(50) and log(2) can be written as log₁₀ 50 and log₁₀ 2. Logarithms whose base is 10 are known as common logarithms.

Another number, ‘e’ also known as Euler’s number (an irrational number equal to 2.71828....) is one of the most used bases in the concept of logarithms. The logarithm of any number n to the base e is written as either ln (n) or as log _e (n) and can be written even excluding the parentheses. Logarithms whose base is e are known as natural logarithms.

History of Logarithm

Scottish physicist and astronomer John Napier came up with the concept of logarithms in the 17^th century. The term logarithm was coined from the Greek words logos and arithmos which mean proportion and number, respectively. Napier then published a book that contained natural logarithms’ log tables. A year later, another mathematician, Englishman Henry Briggs collaborated with Napier and recalibrated the existing log table. They created what is known in the present day as the common logarithms’ log table.
The aim of this finding was to speed up and hence help in long, painstaking calculations that would otherwise take up a lot of time on account of the enormous numbers involved in them. Logarithms helped mathematicians in multiplying gigantic numbers, in raising numbers to higher powers, calculating roots, in problems employing the concept of surds, all before mechanical calculators and computers were invented.

Logarithmic Scales

The notion of logarithmic scales refers to the representation of numeric information employing a rather large range of numbers in a rather narrowed down format. Explained simply, on a logarithmic scale, the concerned values are plotted with reference to the logarithm of the values instead of the actual values. This is different from a linear scale, in which the actual values are depicted as equally distant from each other, as they would occur on a number line.

The numbers 40 and 50, 70 and 80 are not equally spaced from each another, rather the logarithmic scale places 10 and 100, 100 and 1000, 1000 and 10000 at an equal distance from one another. Notice that the difference between these values is not the same, unlike in a linear scale, where the difference between 1 and 2 & 2 and 3 is the same. On a logarithmic scale, rather, the succeeding unit is obtained by multiplying the previous unit by a fixed factor, which in this case is 10. Such scales help to interpret and analyze data spread across a large numerical range by representing it in a more manageable range.

A few of the phenomena in which logarithmic scales are used today are the intensity of earthquakes, brightness of stars, financial growth, population studies, and even decibels (the unit of intensity of sound).

Exponents

The number of times a number n makes an appearance in repeated multiplication is referred to as the exponent of the number n. For instance, 2 x 2 x 2 = 8 or 2³ = 8. Here, 3 is the exponent of 2, since 2 appears 3 times in its own repeated multiplication.

Properties of Logarithm

The exponential equation b^y = x will have its logarithmic form as log_b x = y provided b > 0, where b ≠ 1 and x > 0.

• When b is 1, no matter what the value of y is, it will always yield x to be 1. Because of this, y cannot take a single value but multiple values, and hence remains undetermined.

• When b is 0 and is raised to any power, x will always be zero. As a result of this, y can take numerous values and not a defined one and is undetermined.

• When b is a negative number and y is a fraction, the argument x may or may not be defined. For instance, -64^⅓ or ∛-64 is equal to –4, since (-4)³ will ultimately yield us – 64. However, when we consider (- 64)^½ or ²√-64, the equivalent argument is indeterminate since squaring any number will always yield a positive number and never a negative one. For instance, (-8)² or 8² is always 64. This leads us to conclude that whenever we deal with negative numbers under the radicand symbol, we will find only their n^th root only when the degree of the root is an odd number and never when the degree of the root is an even number.

Because not all negative arguments universally conform to the rules of logarithms, logarithms of negative numbers have not been defined and are confined to only positive arguments.

Rules of logarithms

Product rule	logb xy = logb x + logb y
Quotient rule	logb x/y = logb x – logb y
Power rule	logb xz = z logb x
Identity rule	logb b = 1
Zero rule	logb 1 = 0
Log of exponent rule	logb bp = p
Exponent of log rule	blogb p = p
Change of base rule	1. logb x = loga x / loga b 2. logb x = 1 / logx b 3. logbz x = (1/z)logb x

Logarithm Formulas

• log_b xy = log_b x + log_b y
• log_b x/y = log_b x – log_b y
• log_b x^z = z log_b x
• log_b b = 1
• log_b 1 = 0
• log_b b^p = p
• b^{log_b p} = p
• log_b x = log_a x / log_a b
• log_b x = 1 / log_x b
• log_b^z x = (1/z)log_b x

Logarithm Example:

Q.2) Evaluate:
• log_√6 216
• log_√3 243/81
Solution:
• log_√6 216
= log_√6 6³
= log_√6 6 ^3x2/2
= log_√6 √6⁶
= 6 log_√6 √6 (by power rule)
= 6 x 1(by identity rule)
= 6
• log_√3 243/81
= log_√3 243 - log_√3 81 (by quotient rule)
= log_√3 3⁵ - log_√3 3⁴
= log_√3 3^5x2/2 - log_√3 3^4x2/2
= log_√3 √3¹⁰ - log_√3 √3⁸
= 10 log_√3 √3 - 8 log_√3 √3 (by power rule)
= 10 – 8 (by identity rule)
= 2

Logarithm Example:

Q.3) If p = log_x yz, q = log_yzx and r = log_z xy, find the value of 1/(1+p) + 1/(1+q) + 1/(1+r).
Solution:
1/(1+p) + 1/(1+q) + 1/(1+r)
= 1/(log_x x + log_x yz) + 1/(log_y y +log_y zx) + 1/(log_z z + log_z yx) (by identity rule)
= 1/(log_x xyz) + 1/ (log_y yzx) + 1/(log_z zyx) (by product rule)
= log_xyzx + log_xyz y + log_xyz z (by change of base rule)
= log_xyz xyz
= 1 (by identity rule)

How to use Log Table?

The base of a logarithm can be any positive real number greater than one, 10 and e being two of the most widely used ones. Since the concept of logarithms extensively uses a log table to arrive at values, we will now see how a log table is referred to and put to use. Given below is an excerpt from the common logarithms’ log table in which the base is 10.

The logarithmic form of 10² = 100 is log₁₀ 100 = 2. Here, 2 is known as the characteristic of the logarithm. The logarithmic form of 10³ = 1000 is log₁₀ 1000 = 3. Apparently, the characteristic here is 3. From this, we can conclude that the common logarithm of any number greater than 100 and lesser than 1000 will lie between 2 and 3.
It can be represented with the inequality log₁₀100 < log₁₀N < log₁₀1000, where N can be any number greater than 100 and lesser than 1000.

In the case of logarithms of any positive real numbers bigger than 1, the characteristic is positive and is equal to one less than the number of digits to the left of the decimal point.
On the other hand, in the case of logarithms of any positive real numbers less than one but greater than zero, the characteristic is negative and is one more than the number of zeroes present to the immediate right of the decimal point.
In both cases, the entire part to the right of the decimal point is called the mantissa of the logarithm.
For any number N, log₁₀ N = characteristic + mantissa
For instance, log₁₀ 100 = 2 + 0 = 2

Calculating the logarithm of a number using the common log table:

Following are the steps to be taken to calculate the log of a number, 20.03, using the common log table:

• First, ignore the decimal point. Let us focus on 2003 without thinking about the position of the decimal point.

• Take the first two digits of the number and search for the number formed by the two digits in the leftmost column of the common log table. We look for the number 20 in the leftmost column.

• Next, look for the number that’s located in 20’s row and in the column of the third digit that we consider after we’re done with 20, which is now 0. The number common to 20’s row and 0’s column is 3010.

• Now, we consider the number located in the row of 20 and the column of the fourth digit, which is 3, in the mean difference section of the log table. The number found common is 6.

• Add the numbers found in step 3 and step 4. Hence, 3010 + 6 = 3016. This number will be after the decimal point and will be called mantissa of the log, and is .3016 here.

• To find the characteristic of the log value, we consider the number of digits to the left of the decimal point of the argument, which is 20. Since there are two digits in the number 20, the characteristic will be one less than the number of digits in 20. Hence, the characteristic will be 2-1 = 1.

• log₁₀ N = characteristic + mantissa. Hence, log₁₀ 20.03 = 1 + 0.3016 = 1.3016

Logarithm Example:

Q.4) Calculate the log of 8.056 using the log table.
Solution:
• First, ignore the decimal point. Let us focus on 8056 without thinking about the position of the decimal point.
• Take the first two digits of the number and search for the number formed by the two digits in the leftmost column of the common log table. We look for the number 80 in the leftmost column.
• Next, look for the number that’s located in 80’s row and in the column of the third digit that we consider after we’re done with 80, which is 5. The number common to 80’s row and 5’s column is 9058.
• Now, we consider the number located in the row of 80 and the column of the fourth digit, which is 6, in the mean difference section of the log table. The number found common is 3.
• Add the numbers found in step 3 and step 4. Hence, 9058 + 3 = 9061. This number will be after the decimal point and will be called mantissa of the log, and is 0.9061 here.
• To find the characteristic of the log value, we consider the number of digits to the left of the decimal point of the argument, which is 8. Since there is a single digit in the number 8, the characteristic will be one less than the number of digits in 8. Hence, the characteristic will be 1-1 = 0.
• log₁₀ N = characteristic + mantissa. Hence, log₁₀ 8.056 = 0 + 0.9061 = 0.9061

Logarithm Example:

Q.5) Find the value of log₆₄ 128
Solution:
log₆₄ 128
= (log₁₀ 128) / (log₁₀ 64) (by change of base rule)
= (log₁₀ 2⁷) / (log₁₀ 2⁶ )
= (7 log₁₀ 2) / (6 log₁₀ 2) (by power rule)
= 7/6

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