Notes on Logarithms

Logarithms are another way of writing exponents in mathematics. A number’s logarithm with a base equals another number. Exponentiation is the inverse function of a logarithm. If 102 = 100, for example, log10 100 = 2.

As a result, we may deduce that

bn = x or logb x = n

The base of the logarithmic function is b.

“The logarithm of x to the base b is equal to n,” it says.

We will cover the definition of logarithms, the two types of logarithms (common and natural logarithms), and other properties of logarithms with numerous solved cases in this post.

History

In the 17th century, John Napier introduced the concept of logarithms. Many scientists, navigators, engineers, and others later utilised it to conduct other computations, making it simple. To put it another way, logarithms are the inverse of exponentiation.

What are logarithms, exactly?

The power to which a number must be increased to obtain additional values is defined as a logarithm. It is the most practical method of expressing enormous numbers. Multiplication and division of logarithms can also be stated in the form of logarithms of addition and subtraction, thanks to a number of important features of logarithms.

“The exponent by which b must be raised to give an is the logarithm of a positive real number a with respect to base b, a positive real number not equal to 1[nb 1].”

in other words, by = a⇔ logba=y

Where,

The numbers “a” and “b” are both positive real numbers.

y is an integer.

“a” stands for argument and is located inside the log. “b” stands for base and is located at the bottom of the log.

In other terms, the logarithm answers the question, “How many times does a number have to be multiplied to get the other number?”

For instance, how many threes must be multiplied to get the answer 27?

The answer is 27 if we multiply three times.

As a result, the logarithm is three.

The following is the logarithm form:

Log3 (27) = 3 …. (1)

As a result, 27’s base 3 logarithm is 3.

The logarithm form shown above can also be written as:

3x3x3 = 27

33 = 27 ….. (2)

As a result, equations (1) and (2) have the same meaning.

Types of Logarithms

We always deal with two forms of logarithms in most circumstances, notably

Logarithm (common)

Logarithm (natural)

Logarithm (common)

The base 10 logarithms are also known as the common logarithm. It is written as log10 or just log. The common logarithm of 1000, for example, is expressed as a log (1000). The common logarithm tells us how many times we must multiply 10 to reach the desired result.

For instance, log(100) = 2

The result of multiplying the number 10 twice is 100.

Logarithm (natural)

The base e logarithm is the natural logarithm. ln or loge is the symbol for the natural logarithm. The Euler’s constant, which is approximately equal to 2.71828, is represented by “e.” The natural logarithm of 78, for example, is written as ln 78. The natural logarithm indicates how many times we must multiply “e” to obtain the desired result.

ln (78) = 4.357, for example.

As a result, the base e logarithm of 78 is 4.357.

Logarithm Properties and Rules

Logarithmic operations can be conducted according to a set of rules. These rules are known as:

Product law

Rule of division

Rule of Power/Exponential Rule

Base rule modification

Log’s derivative

Let’s take a closer look at each of these characteristics one by one.

Product Law

According to this rule, multiplying two logarithmic numbers equals adding their individual logarithms.

Logb (mn) = logb m + logb n

For instance, log3 (2y) = log3 (2) + log3 (2) + log3(2) (y)

Rule of Division

The difference of each logarithm equals the division of two logarithmic numbers.

logb m – logb n

For instance, log 3(2/y) = log3 (2) -log3 (2) (y)

Rule of Exponentials

The logarithm of m with a rational exponent is equal to the exponent times its logarithm, according to the exponential rule.

n logb m = logb (mn)

For instance, logb(23) = 3 logb2

Base Rule Modification

loga m/ loga b = logb m

For instance, logb 2 = loga 2/loga b

Logb (a) = 1 / loga is the base switch rule (b)

For instance, logb 8 = 1/log8 b

Log’s Derivative

If f (x) = logb (x), the derivative of f(x) is as follows:

f'(x) = 1/(x ln(b))

Assume that f (x) = log10 (x)

f'(x) = 1/(x ln(10)) then

x(logb(x) – 1/ln(b))dx = x(logb(x) – 1/ln(b)) + C

Example: (log10(x) – 1 / ln(10)) + C log10(x) dx = x

Examples of Logarithms 

Example 1:

Calculate log 2 (64) =?

Solution:

Because 26= 2 *2*2*2*2*2= 64, the exponent value is 6, and log 2 (64)= 6.

Example 2:

What is the log10(100) value?

Solution:

In this instance, 102 = 100. As a result, 2 is the exponent value, and log10(100)= 2

Example 3:

Using the logarithm property, find the value of x for log3 x= log3 4+ log3 7

Solution:

Log3 4+ log3 7= log 3 (4 * 7) according to the addition rule

Log 3( 28 ). Thus, x= 28.

Conclusion

In today’s world of research and technology, logarithms are commonly employed. We can even discover logarithmic calculators, which have greatly simplified our calculations. Surveying and celestial navigation are two applications for them. They’re also used in computations like determining the loudness (decibels), the earthquake’s intensity on the Richter scale, radioactive decay, acidity (pH= -log10[H+]), and so on.