Laws of Logarithms

The logarithm is the exponent or power to which a base is raised to get a particular number. For example, ‘a’ is the logarithm of ‘m’ to the base of ‘x’ if x^m = a, then we can write it as m = log_xa. Logarithms are invented to speed up the calculations and time will be reduced when we are multiplying many digits using logarithms. Now, Let’s discuss the laws of logarithms below.

Laws of Logarithms

There are three laws of logarithms that are derived using the basic rules of exponents. The laws are the product rule law, quotient rule law, power rule law. Let’s take a look at the laws in detail.

First Law of logarithm or Product Rule Law

Let a = xⁿ and b = x^m where base x should be greater than zero and x is not equal to zero. i.e., x > 0 and x ≠ 0. from this we can write them as

n = log_xa and m = log_xb ⇢ (1)

By using the first law of exponents we know that xⁿ × x^m = x^{n + m} ⇢ (2)

Now we multiply a and b we get it as,

ab = xⁿ × x^m

ab = x^{n + m} (From equation 2)

Now apply the logarithm to the above equation we get as below,

log_xab = n + m

From equation 1 we can write as log_xab = log_xa + log_xb

So, if we want to multiply two numbers and find the logarithm of the product, then add the individual logarithms of the two numbers. This is the first law of Logarithms/ Product Rule Law.

log_xab = log_xa + log_xb

We can apply this law for more than two numbers i.e.,

log_xabc = log_xa + log_xb + log_xc.

Second Law of logarithm or Quotient Rule Law

Let a = xⁿ and b = x^m where base x should be greater than zero and x is not equal to zero. i.e., x > 0 and x ≠ 0. from this we can write them as,

n = log_xa and m = log_xb ⇢ (1)

By using the first law of exponents we know that xⁿ / x^m = x^{n – m} ⇢ (2)

Now we multiply a and b we get it as,

a/b = xⁿ / x^m

a/b = x^{n – m} ⇢ (From equation 2)

Now apply the logarithm to the above equation we get as below,

log_x(a/b) = n – m

From equation 1 we can write as log_x(a/b) = log_xa – log_xb

So, if we want to divide two numbers and find the logarithm of the division, then we can subtract the individual logarithms of the two numbers. This is the second law of Logarithms/ quotient Rule Law.

log_x(a/b) = log_xa – log_xb

Third Law of logarithm or Power Rule law

Let a = xⁿ ⇢ (i),    

Where base x should be greater than zero and x is not equal to zero. i.e., x > 0 and x ≠ 0. from this we can write them as,

n = log_xa ⇢ (1)

If we raise both sides of the equation(i) with the power of ‘m’ then we get it as follows,

a^m = (xⁿ)^m = x^nm

Let a^m be a single quantity and apply logarithm to the above equation then,

log_xa^m = nm

log_xa^m = m.log_xa

This is the third law of logarithms. It states that the logarithm of a power number can be obtained by multiplying the logarithm of the number by that number.

Sample Problems

Problem 1: Expand log 21.

Solution: 

As we know that log_xab = log_xa + log_xb (From first law of logarithm)

So, log 21 = log (3 × 7) 

= log 3 + log 7

Problem 2: Expand log (125/64).

Solution: 

As we know that log_x(a/b) = log_xa – log_xb (From second law of logarithm)

So, log (125/64) = log 125 – log 64

= log 5³ – log 4³

log_xa^m = m.log_xa (From third law of logarithm), we can write it as,

= 3 log 5 – 3 log 4

= 3(log 5 – log 4)

Problem 3: Write 3log 2 + 5 log3 – 5log 2 as a single logarithm.

Solution: 

3log 2 + 5 log3 – 5log 2

= log 2³ + log 3⁵ – log 2⁵

= log 8 + log 243 – log 32

= log(8 × 243) – log 32

= log 1944 – log 32

= log (1944/32)

Problem 4: Write log 16 – log 2 as a single logarithm.

Solution:

log(16/2)

= log(8)

= log(2³)

= 3 log 2

Problem 5: write 3 log 4 as a single logarithm

Solution: 

From the power rule law, we can write it as,

= log 4³

= log 64

Problem 6: Write 2 log 3- 3 log 2 as a single logarithm

Solution: 

log 3² – log 2³

= log 9 – log 8 

= log (9/8)

Problem 7: Write log 243 + log 1 as a single logarithm

Solution: 

log (243 × 1) 

= log 243