where Ω ⊂ R N is an open bounded set, α , β ∈ R , N ≥ 3 , μ ∈ R + be such that 0 < μ < ( N − 2 ) 2 / 4 , and 2 * = 2 N / ( N − 2 ) is the critical Sobolev exponent. The novelty of our work lies in investigation of crucial energy estimates involving logarithmic terms, which are essential to overcome the lack of compactness. With the help of these estimates, we obtain our results in two cases for α ∈ R . In the first case, that is, β > 0 and μ ¯ − μ ≥ 1 , we prove the existence of a positive ground state solution. While in the second case, for β < 0 and μ ¯ − μ < 1 , we show that the above problem admits a positive solution under certain assumptions on α , β , and N . Further, we obtain the nonexistence result of a positive solution for β < 0 . Moreover, compared to the above problem without the logarithmic term, our results hold for a wider range of α . To the best of our knowledge, no existing result has been studied so far for critical elliptic equations involving Hardy potential and logarithmic perturbation.