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 logx 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: am.an=am+n
  • Quotient rule: am/an = am-n
  • Power of a Power: (am)n = amn

Now let us learn the properties of the logarithm.

Product Property

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

loga(mn) = logam + logan

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: logb bx = x)

= 5

Quotient Property

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

loga(m/n) = logam – logan

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

Example: log2(21/8)

log2(21/8) = log21 – log8

Power rule

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

logamn = n logam

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 one for logarithms. Some other properties are:

Change of Base rule

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

Logn m = logp m/logp n

Example: 

log2 10 = logp 10/logp 2

Reciprocal rule

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

logn m = 1/logmn

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:

Properties/Rules Exponents Logarithms
Product Rule xp.xq = xp+q loga(mn) = logam + logan
Quotient Rule xp/xq = xp-q loga(m/n) = logam – logan
Power Rule (xp)q = xpq logamn = n logam

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 pq = 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.

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