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We use the log-likelihood because the natural logarithm is a strictly increasing (monotonic) function, so maximizing the log-likelihood yields the exact same parameter value as maximizing the original likelihood. Additionally, the logarithm converts the product of independent probabilities into a sum, which makes differentiation much easier.

Conditions: The likelihood function $L(\theta)$ is strictly positive.; The base of the logarithm is greater than 1 (e.g., the natural logarithm).

We use the log-likelihood because the natural logarithm is a strictly increasing (monotonic) function, so maximizing the log-likelihood yields the exact same parameter value as maximizing the original likelihood. Additionally, the logarithm converts the product of independent probabilities into a sum, which makes differentiation much easier.

Conditions: The likelihood function $L(\theta)$ is strictly positive.; The base of the logarithm is greater than 1 (e.g., the natural logarithm).