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## Main Question or Discussion Point

I'm learning complex analysis right now, and I'm reading from Joseph Taylor's Complex Variables.

On Theorem 1.4.8, it says "If a log is the branch of the log function determined by an interval I, then log agrees with the ordinary natural log function on the positive real numbers if and only if the interval I contains 0."

Can anyone help me understand what this means? Is it just saying that the properties of the real log function only hold if the interval I contains 0? I don't understand why.

On Theorem 1.4.8, it says "If a log is the branch of the log function determined by an interval I, then log agrees with the ordinary natural log function on the positive real numbers if and only if the interval I contains 0."

Can anyone help me understand what this means? Is it just saying that the properties of the real log function only hold if the interval I contains 0? I don't understand why.