Logarithms

Logarithm is a way of writing numbers which are in exponential form. In Exponential Form, a number is raised to some power, and the value is calculated by multiplying the number by itself to the number of times equal to the power raised on it. In Logarithm we try to find what power a number is raised that resulted in a particular number. Let’s say we need to find 4³ we find it as 4 × 4 × 4 = 64 by the basics of Exponents that we have learned. But what if we need to find out what power is four is raised so that the result is 64? In this case, we need to find out by hit and trial but this technique will fail for large numbers, hence we need to solve by logarithm log₄64 = 3. 

Thus, we write 4³ = 64 as log₄64 = 3. In general terms, we can write it as aⁿ = b or log_ab = n.

In this article, we will learn, about the history, and properties of logarithms along with techniques to solve questions also we will learn about logarithmic functions, their curve, differentiation, and integration in detail and examples.

History of Logarithms

The creation of logarithms is credited to a Scottish Mathematician John Napier. In 1614 he presented his book named Canons of Logarithms which contained a table of trigonometric functions and their natural logarithms. The purpose of the book was to help in the multiplication of quantities. The creation of Logarithms is considered one of the important scientific discoveries in the history of science that were helpful in simplifying calculations of spherical trigonometry and celestial navigation.

What is Logarithm?

Logarithms are a way of determining the power to which a number is raised that gives a particular number. In simpler words, we can say that if a and b are two numbers such that ‘a’ is raised to some power that gives the number ‘b’ then logarithm is used to find what is the power that ‘a’ must be raised to yield ‘b’. Logarithm is often referred to as the inverse process of exponents.

Log Definition

If aⁿ = b then log or logarithm is defined as the log of b at base a is equal to n. It should be noted that in both cases base is ‘a’ but in the log, the base is with the result and not the power.

aⁿ = b ⇒ log_ab = n

where, 

• a is the base
• b is the argument
• a and b positive real numbers
• n is real number

Exponential to Log Form

If a number is expressed in the exponential form for example an = b where a is the base, n is the exponent and b is the result of the exponent then to convert it into the logarithmic form the base ‘a’ remains base in logarithm, the result ‘b’ becomes an argument and the exponent ‘n’ becomes the result here.

aⁿ = b ⇒ log_ab = n

Log to Exponential Form

If the expression is in logarithmic form then we can convert it into exponential form by making the argument as a result and the result of logarithm becomes the exponent while the base remains the same. It can be better understood from the expression mentioned below:

log_ab = n ⇒ aⁿ = b

Logarithm Types

Depending upon the base, there are two types of logarithm

• Common Logarithm
• Natural Logarithm

1. Common Logarithm

The logarithm with base 10 is known as Common Logarithm. It is written as log₁₀X. The common logarithm is generally written as log only instead of log₁₀.

Let’s see some example

• Log10 = log₁₀10 = 1
• log1 = log₁₀1 = 0
• log1000 = log₁₀1000 = 3

2. Natural Logarithm

The logarithm with base e, where e is a mathematical constant is called Natural Logarithm. It is written as log_eX. The natural logarithm is also written in the abbreviated form as ln i.e. log_eX = ln X.

Let’s see some example:

• log_e2 = X ⇒ e^X = 2
• log_e5 = y ⇒ e^y = 5

Difference between Log and ln

The basic difference between log and ln is tabulated below:

Related Read: Difference Between Log and Ln

Logarithm Rules and Properties

The common Properties of Log are mentioned below:

• Product Rule
• Division Rule
• Power Rule
• Change of Base Rule
• Base Switch Rule
• Equality of Log
• Number raised to Log Power
• Negative Log Rule

1. Product Rule of Log

Product rule of log states that if log is applied to the product of two numbers then it is equal to the sum of the individual logarithmic values of the numbers. The expression can be given as:

log_xab = log_xa + log_xb

Example: loga10 = loga(5 ✕ 2) = loga5 + loga2

2. Quotient Rule of Log

Quotient Rule of log states that if log is applied to the quotient of two numbers then it is equal to the difference of the individual logarithmic value of the numbers. The expression can be given as:

log_xa/b = log_xa – log_xb

Example: log_x2 = log_x(10/5) = log_x10 – log_x5

3. Power Rule of Log

Power Rule of log states that if the argument is raised to some power then the solution of logarithmic expression is given by the power of the argument multiplied by the log value of the argument. The expression can be given as:

log_xa^b = b.log_xa

4. Change of Base Rule

In logarithm, the base can be changed in the following way

log_xa = log_ya/log_yb

OR

log_xa.log_yb = log_ya

Related Read: Change of Base Rule

5. Base Switch Rule

This log property states that Base and argument can be switched in the following manner

log_xa = 1/log_ax

6. Equality of Logarithm

Equality of logarithm property states that if

log_xa = log_xb then a = b

7. Number Raised to Log

If a number is raised to log which has the same base as the number then the result of the expression is the argument. This can be expressed as

x^log_x^a = a

8. Negative Log Rule

The Negative Log Property states that if the logarithmic expression is of the form -log_xa then we can convert it into a positive form by taking the reciprocal of the argument or by taking the reciprocal of the base as

-log_xa = log_x(1/a) = log_1/xa

Related Read: Laws of Logarithm

Some Other Properties of Log

Apart from the above-mentioned properties, there are some other properties of Log. Using these properties we can directly put their values in any equation. These properties are mentioned below:

1. Log 1

This property of log states that the value of Log 1 is always zero, no matter what the base is. This is because any number raised to power zero is 1. Hence, Log 1 = 0.

2. Log_aa

This Property of Log states that if the base and augment of a logarithm are the same then the logarithm of that number is 1. This is because any number raised to power 1 results in the number itself. Hence,

ln e = log_ee = 1

log₁₀10 = 1

log₂2 = 1

3. Log 0

This rule states that the log of zero is not defined as there is no such number when raised to any power that results in zero. Hence, log 0 = Not defined.

Expanding and Condensing Logarithm

A logarithmic expression can be expanded and condensed using the following Properties of Log

• Product Rule of Log: log_xab = log_xa + log_xb
• Quotient Rule of Log: log_xa/b = log_xa – log_xb
• Power Rule of Log: log_xab = b.log_xa

1. Expanding Log

Log can be expanded in the following manner.

Example: Expand log(2a³b²)

Solution:

log(2a³b²) = log 2 + log a³ + log b²

= log 2 + 3 log a + 2 log b

2. Condensing Log

A log can be condensed in the following manner just by following the reverse of the properties of log.

Example: Condense log 2 + 3 log a + 2 log b

Solution:

log 2 + 3 log a + 2 log b

= log 2 + log a3 + log b2

= 2a³b²

Logarithmic Formulas

The formulas for logarithm is tabulated below:

Related Read: Logarithm Formulas

Logarithmic Equations

An equation that involves logarithmic expression equated to any number or other logarithmic term is called Logarithmic Equation. In Logarithmic Equation we assume that the base of all the logarithmic terms is the same if no specific base is mentioned. In a Logarithmic Equation if the base of all the log terms is the same the calculation becomes easy as compared to the situation when the base of log terms is different. To solve a Logarithmic Equation we need to remember some of the facts mentioned below:

If log_ax = log_ay ⇒ x = y

If y = log_ax ⇒ a^y = x

Let’s see some examples based on it:

Example 1: Find x for ln 6 – ln(5 – x) = ln x

Solution:

Using Quotient Rule

ln{6/(5 – x)} = ln x

⇒ 6/(5 – x) = x

⇒ 5x -x² = 6

⇒ x² – 5x + 6 = 0

⇒ x² – 3x – 2x + 6 = 0

⇒ x(x – 3) – 2(x – 3) = 0

⇒ (x – 2)(x – 3) = 0

⇒ x = 2 or x = 3

Example 2: Find x for log₃(3x + 6) = 3

Solution:

Using the formula, If y = log_ax ⇒ a^y = x

(3x + 6) = 3³

⇒ 3x = 27 – 6

⇒ x = 21/3 = 7

Log Table

Log Table is used to find the value of the log without the use of a calculator. The log table provides the logarithmic value of a number at a particular base. A log table has mainly three columns. The first column contains two-digit numbers from 10 to 99, the second column contains differences for digits 0 to 9 and hence called the difference column and the third column contains mean difference from 1 to 9 and hence called mean difference column. The log table for base e is called the natural logarithm table and that for base 2 is called the binary log table.

The logarithmic value of a number contains two parts named characteristics and mantissa both separated by a decimal. Characteristic is the integral part written o the left side of the table and can be positive or negative while the mantissa is the fraction or decimal that is always positive. One needs to find characteristics and mantissa from the log table and combine them to get the logarithmic values. 

Anti Log Table

Antilog is the process of finding the inverse of the log of the number. This is used when the number is already given in log value and we need to find out the number for which log value is given. If log a = b then a = antilog (b). The Antilog table is helpful in finding the Antilog value without using the calculator. The antilog table also consists of 3 columns among which the first column contains numbers from .00 to .99, the second block which is the difference column contains digits from 0 to 9, and the third block which is the mean difference column contains digits from 1 to 9.

Using Log Table we have find out the characteristics and mantissa of a number while using the antilog table we will separate the characteristics and mantissa of the number.

Logarithmic Function

A function that is given in the form of y = f(x) = log_ax where a > 0 is called the logarithmic function. It is the inverse of the exponential function given as a^y = x. The logarithmic function includes both natural and common logarithms.

Domain and Range of Logarithmic Function

Domain of a function is the input value for which it gives an output. The output obtained for the set of domains in which the function is defined is called the Range of the function. We have learned that we find the log only of a non-negative number, hence the domain of the logarithmic function is a set of all non-negative real numbers (0, ∞). The logarithmic function is both continuous and differentiable in its domain. We also observed that the logarithmic value of a logarithmic function is both positive and negative numbers. Hence, the range of the logarithmic function is a set of all Real Numbers i.e. (R).

Logarithmic Graph

We know that the domain of Logarithmic Function is (0, ∞) and its range is a set of all real numbers. If we plot the graph using the set of domain and range we find that the graph of the logarithmic function is just the inverse of the graph obtained for the exponential function. This indicates the inverse relationship between exponential and logarithmic functions. Also, the logarithmic graph is symmetric around the line y = x. We know that the value of log 1 is zero at any base value. Hence it has an intercept (1,0) on the x-axis and no intercept on the y-axis as log 0 is not defined.

Logarithmic Graph

Properties of Logarithmic Graph

There are the following properties of the logarithmic Graph of function log_ax

• In logarithmic function base a > 0 and a ≠ 1
• The graph of logarithmic function increases when a > 1 and decreases in the range 0 < a < 1.
• The domain of the function is a set of all positive numbers greater than zero.
• The range of the curve is a set of all real numbers.

Properties of Logarithmic Function

The logarithmic function has the following properties

• log xy = log x + log y
• log x/y = log x – log y
• log ab = log_cb/log_ca
• log 1 = 0
• log 0 = Not defined
• log_aa = 1

Limit of Logarithmic Function

The fundamental formula for the limit of the logarithmic function is mentioned below:

• Iim_x→0 log_ex = 1
• Iim_x→0 log_e(1 + x)/x = 1
• Iim_x→0 log_b(1 + x)/x = 1/log_eb = log_be
• Iim_x→0 log_e(1 – x)/x = -1

Derivative of Log

Derivative refers to the change in one quantity let’s say ‘y’ with respect to another quantity let’s say ‘x’. The derivative of a function y = f(x) is given by f'(x) = dy/dx. Below are some standard derivative formulas that involve the use of Euler’s number or a logarithmic function.

• d/dx{e^x} = e^x
• d/dx(a^x) = a^x.log_ea | where a > 0 and a ≠ 1
• d/dx(log_ex) = 1/x, x > 0
• d/dx(log_ax) = 1/x(log_ae) = 1/xlog_ea | where x > 0

Logarithmic Differentiation

Logarithmic Differentiation is used to find the derivative of a composite function using the concept of logarithm. In order to differentiate a composite function with the technique of logarithmic differentiation, first simplify the function using log formulas and then differentiate it using the chain rule. Derivative of some composite functions using logarithmic differentiation is mentioned below:

The following articles should be referred to for details of the differentiation of the logarithmic function:

• Logarithmic Differentiation
• Continuity and Differentiability of Logarithm
• Derivative of Exponential and Logarithmic Functions

Integral of Log

The integration of log x gives the area under the logarithmic curve. Here we integrate the natural log and not the common log i.e. the log with base e. The integration of a logarithmic function is done by the technique of integration by parts. The priority order of taking functions as first and second is decided using the ILATE rule. Here, the logarithmic function is second in priority order. The formula for indefinite and definite integration of log_ex or ln x is given below:

• Indefinite Integral of log_ex: ∫log_ex dx = ∫ln x dx = x.lnx – x + C | where C is constant
• Definite Integral of log_ex: ln x dx = -1

Logarithm Applications

Logarithm is an important concept to solve problems in physics, chemistry, and mathematics. Logarithm is used to solve problems of pH and radioactive decay in chemistry. In Physics, the concept of log is used to solve problems based on the loudness of sound, celestial navigation, earthquake intensity, and many more. In mathematics, the concept of logarithm is used to solve larger powers in an easy manner.

Also Read,

• Calculus
• Algebra
• Probability

Logarithm Examples

Example 1: Find log_a16 + 1/2 log_a225 – 2log_a2

Solution:

log_a16 + 1/2 ✕ 2log_a15 – log_a2²

⇒ log_a16 + log_a15 – log_a4

⇒ log_a(16 ✕ 15) – log_a4

⇒ log_a(16 ✕ 15/4) = log_a60

Example 2: Solve log_b3 – log_b27

Solution:

log₂3 – log₂48

⇒ log₂(3/48)

⇒ log₂(1/16)

⇒ log₂(-16)

⇒ -log₂2⁴

⇒ -4log₂2 = -4

Example 3: Find x in log_bx + log_b(x – 3) = log_b10

Solution:

Given log_bx + log_b(x – 3) = log_b10

⇒ log_b(x)(x – 3) = log_b10

⇒ (x)(x – 3) = 10

⇒ x² – 3x – 10 = 0

⇒ x² – 5x + 2x – 10 = 0

⇒ x(x – 5) + 2(x – 5)

⇒ (x – 5)(x + 2) = 0

⇒ x = 5, -2

Example 4: Find the derivative of log(sin x)

Solution:

Given that y = log(sin x)

dy/dx = (1/sin x) × cos x = cos x/sin x = tan x

Logarithms – FAQs

1. What are Logarithms?

Logarithm is the inverse of exponential which is used to find out to what power a base must be raised to yield a particular value.

2. Who invented Logarithm?

Logarithm was invented by Scottish Mathematician John Napier in 1614.

3. Can Logarithm be Negative?

No the argument of logarithm can’t be negative, however, log of any number can give a negative value.

4. What are Logarithms used for?

Logarithm is used to find the power a number must be raised to yield a particular result. It is useful in finding pH level, exponential growth or decay, etc.

5. What is the value of log_a0?

Logarithm does not take 0 and negative numbers as its input hence loga0 is not defined.