State whether each of the following statements is true or false.

(i) log₅125 = 3

(ii)

(iii) log₄(6 + 3) = log₄6 + log₄3

(iv)

(v)

(vi) log_aM - N = log_aMlog_aN

(i) True

log₅125 = 3

⇒ 5³ = 125

(∵ x = log_ab is the logarithmic form of the exponential form a^x = b)

This is true.

(ii) False

⇒

(∵ x = log_ab is the logarithmic form of the exponential form ax = b)

Here

Therefore, this False.

(iii) False

Here its given log₄(6 + 3) = log₄6 + log₄3

Let us consider the RHS i.e.

log₄6 + log₄3 = log₄(6×3) (∵ according to the product rule loga(M×N) = logaM + logaN;

a,M,N are positive numbers,a≠1)

But here LHS is log4 (6 + 3)

Hence it’s False.

(iv) False

Here it’s given

Let us consider the LHS i.e.

(∵ log_aM ÷ log_aN = log_aM - log_aN

;a,M,N are positive numbers ,a≠1)

But here the RHS is

Hence both the sides are not equal and therefore it’s False.

(v) True

Here it’s given:

⇒ (∵ x = log_ab is the logarithmic form of the exponential form a^x = b)

⇒

Hence LHS = RHS

Therefore this is True.

(vi) False

Here it’s given that log_a (M - N) = log_a M ÷ log_aN

Let us consider the RHS

log_aM ÷ log_aN = log_aM - log_aN

(∵ according to quotient rule,log_aM ÷ log_aN = log_aM - log_aN ;a,M,N are positive numbers,a≠1)

But the LHS is log_a(M - N)

Therefore LHS≠RHS

Hence it’s False.