Properties of Logarithms

In Mathematics, properties of logarithms functions are used to solve logarithm problems. We have learned many properties in basic maths such as commutative, associative and distributive, which are applicable for algebra. In the case of logarithmic functions, there are basically five properties.

The logarithmic number is associated with exponent and power, such that if xⁿ = m, then it is equal to log_x m=n. Hence, it is necessary that we should also learn exponent law. For example, the logarithm of 10000 to base 10 is 4, because 4 is the power to which ten must be raised to produce 10000: 10⁴ = 10000, so log₁₀10000 = 4.

With the help of these properties, we can express the logarithm of a product as a sum of logarithms, the log of the quotient as a difference of log and log of power as a product.

Only positive real numbers have real number logarithms, negative and complex numbers have complex logarithms.

Logarithm Base Properties

Before we proceed ahead for logarithm properties, we need to revise the law of exponents, so that we can compare the properties.

For exponents, the laws are:

• Product rule: a^m.aⁿ=a^m+n
• Quotient rule: a^m/aⁿ = a^m-n
• Power of a Power: (a^m)ⁿ = a^mn

Now let us learn the properties of logarithmic functions.

Product Property

If a, m and n are positive integers and a ≠ 1, then;

log_a(mn) = log_am + log_an

Thus, the log of two numbers m and n, with base ‘a’  is equal to the sum of log m and log n with the same base ‘a’.

Example: log₃(9.25)

= log₃(9) + log₃(27)

= log₃(3²) + log₃(3³)

= 2 + 3 (By property: log_b b^x = x)

= 5

Quotient Property

If m, n and a are positive integers and a ≠ 1, then;

log_a(m/n) = log_am – log_an

In the above expression, the logarithm of a quotient of two positive numbers m and n results in a difference of log of m and log n with the same base ‘a’.

Example: log₂(21/8)

log₂(21/8) = log₂ 21 – log₂ 8

Power rule

If a and m are positive numbers, a ≠ 1 and n is a real number, then;

log_amⁿ = n log_am

The above property defines that logarithm of a positive number m to the power n is equal to the product of n and log of m.

Example:

log₂10³ = 3 log₂10

The above three properties are the important ones for logarithms. Some other properties are given below along with suitable examples.

Change of Base rule

If m, n and p are positive numbers and n ≠ 1, p ≠ 1, then;

Log_n m = log_p m/log_p n

Example: 

log₂ 10 = log_p 10/log_p 2

Reciprocal rule

If m and n are the positive numbers other than 1, then;

log_n m = 1/log_mn

Example:

log₂ 10 = 1/log₁₀ 2

Also, read:

Comparison of Exponent law and Logarithm law

As you can see these log properties are very much similar to laws of exponents. Let us compare here both the properties using a table:

Natural Logarithm Properties

The natural log (ln) follows the same properties as the base logarithms do.

• ln(pq) = ln p + ln q
• ln(p/q) = ln p – ln q
• ln p^q = q log p

Applications of Logarithms

The application of logarithms is enormous inside as well as outside the mathematics subject. Let us discuss brief description of common applications of logarithms in our real life :

• They are used for the calculation of the magnitude of the earthquake.
• Logarithms are being utilized in finding the level of noise in terms of decibels, such as a sound made by a bell.
• In chemistry, the logarithms are applied in order to find acidity or pH level.
• They are used in finding money growth on a certain rate of interest.
• Logarithms are widely used for measuring the time taken by something to decay or grow exponentially, such as bacteria growth, radioactive decay, etc.
• They can also be used in the calculations where multiplication has to be turned into addition or vice versa.

To learn more about logarithms and other functions, visit byjus.com also, download BYJU’S – The Learning App to learn the maths concepts interactively.