Introduction:

Scottish mathematician John Napier (1550–1617), who introduced the word from the Greek terms for ratio (logos) and number (arithmos) in the 17th century, invented logarithms as a calculation device. To extend a single logarithm into several logarithms or reduce multiple logarithms into a single logarithm, the logarithm properties are applied. A logarithm is simply a different way of expressing exponents. As a result, logarithm properties are derived from exponent attributes. To know the logarithms properties, we should first have a basic idea about what logarithms are. Here, we have study material notes on logarithms properties. 

Definition of logarithms:

A logarithm is a mathematical procedure that defines how many times a given number, known as the base, is multiplied by itself to arrive at another number. The logarithm is described as the inverse function of exponentiation. The logarithm of a specific number x is the exponent to which the other fixed number, the base b, must be increased to obtain that number x. The logarithm records the number of times the same factor appears in repeated multiplication in the most straightforward instance.

The formal definition of logarithms:

The exponent by which b must be raised to get x is the logarithm of a positive actual number x about base b. In other words, the unique real number y is the logarithm of x to base b such that by= x.

” logbx, base b, of x” is the notation for the logarithm.

Laws of logarithms:

There are four laws of logarithms, they are:

• Product Rule Law: The logarithm of a product is equal to the sum of logarithms, according to the product rule. We can add the exponents because logs are exponents and we multiply like bases.

loga (MN) = loga M + loga N

• Quotient Rule Law: The logarithm of a quotient is equivalent to a difference of logarithms, according to the quotient rule. The inverse property can be used to generate the quotient rule, just like the product rule.

loga (M⁄N) = loga M – loga N

• Power Rule Law: By expressing the logarithm of power as the product of the exponent times the logarithm of the base, the power rule for logarithms can be used to simplify it.

IogaMn = n Ioga M

• Change of Base Rule Law: The change of base formula is a formula that uses just log computations with a base of 10 to give you the answer of a log with another base.

loga M = logb M × loga b

Logarithm Properties

The rules of logarithms, derived from the exponent rules, are the properties of the log. These logarithmic properties are used to simplify logarithmic statements and solve logarithmic problems.

Below are some logarithm properties:

• Natural Log Properties:

The natural logarithm is simply a logarithm with base “e” namely, loge = ln. All of the above properties are expressed in terms of “log” and apply to any base; thus, all of the above properties apply to natural logs as well.

The natural logarithm properties are:

•  ln 1 = 0
• ln e = 1
• ln (mn) = ln m + ln n
• ln (m/n) = ln m – ln n
• ln mn = n ln m
• eln x = x

• Product property of log:

The sum of logs is used to represent the logarithm of a product using the product property of logarithms.

• Quotient property of log:

The difference of logs is used to express the logarithm of a quotient using the quotient property of logarithms. By rewriting a logarithm or a quotient as the difference of individual logarithms, the quotient rule for logarithms can be used to compress it.

log​b​​(​NM)=log​b​​M−log​b​​N

1. Power property of logarithms: By expressing the logarithm of a power as the product of the exponent and the logarithm of the base, the power rule for logarithms can be used to simplify it.

logb(Mn)=nlogbM

2.Change of Base property of log: The change of base formula is a formula that uses just log computations with a base of 10 to give you the answer of a log with a different base.

logb(x)=log10(x)log10(b)

Points to remember in logarithms properties:

• The logarithmic properties apply to any log, regardless of base. i.e., they can be used with log, ln, (or) loga.
• The following are the three most important properties of logarithms:

log mn = log m + log n
log (m/n) = log m – log n
log mn = n log m

• Regardless of the bases, log 1 = 0.

To expand or compress logarithms, logarithmic characteristics are used.

Conclusion:

Logarithms properties can be of the very significance of the. We may rewrite the log of a product as a sum of logarithms using the product rule of logarithms. To rephrase the log of a quotient as a difference of logarithms, we can utilise the quotient rule of logarithms. We may rewrite the log of power as the product of the exponent and the log of its base using the power rule for logarithms. Sums, differences, and products with the same base as a single logarithm can also be condensed using logarithm principles.