Consider the following statements :

1. log₁₀ 50 is a rational number.

2. log₁₀₀ 10 is an irrational number.

Which of the statements given above is/are correct ?

Concept:

• For any base b which is a transcendental number the logarithm log_bn of every integer n ≥ 2 is an irrational number.
• A rational number is a number, which can be written as p/q where p & q are integers and q ≠ 0. 

Formula used:

• log mn = log m + log n 
• log_aa = 1
• log_aⁿb = (1/n) log_ab

Calculation:

Statement:1 log₁₀50 is a rational number.

log1050 = log1050 =  log10(5 × 10)

log1050 = log105 + log1010   

log1050 = log105 + 1

As discussed, log105 is an irrational number, log1050 will be also irrational.

Statement:2  log100 10 is an irrational number.

log₁₀₀10 = log_10²10 = (1/2)log₁₀10 = 1/2

Which is a rational number.

∴ Both statements are incorrect.