Summary: In this work, the notion of a “logarithmically absolutely monotonic function” is introduced, the inclusion that a logarithmically absolutely monotonic function is also absolutely monotonic is revealed, the logarithmically complete monotonicity and the logarithmically absolute monotonicity of the function \((1+\alpha/x)^{x+\beta}\) are proved, where \(\alpha\) and \(\beta\) are given real parameters. A new proof for the inclusion that a logarithmically completely monotonic function is also completely monotonic is given, and an open problem is posed.