Blind reduction of speckle noise has become a long-standing unsolved problem in several imaging applications, such as medical ultrasound imaging, synthetic aperture radar (SAR) imaging, and underwater sonar imaging, etc. The unwanted noise could lead to negative effects on the reliable detection and recognition of objects of interest. From a statistical point of view, speckle noise could be assumed to be multiplicative, significantly different from the common additive Gaussian noise. The purpose of this study is to blindly reduce the speckle noise under non-ideal imaging conditions. The multiplicative relationship between latent sharp image and random noise will be first converted into an additive version through a logarithmic transformation. To promote imaging performance, we introduced the feature pyramid network (FPN) and atrous spatial pyramid pooling (ASPP), contributing to a more powerful deep blind DeSPeckling Network (named as DSPNet). In particular, DSPNet is mainly composed of two subnetworks, i.e., Log-NENet (i.e., noise estimation network in logarithmic domain) and Log-DNNet (i.e., denoising network in logarithmic domain). Log-NENet and Log-DNNet are, respectively, proposed to estimate noise level map and reduce random noise in logarithmic domain. The multi-scale mixed loss function is further proposed to improve the robust generalization of DSPNet. The proposed deep blind despeckling network is capable of reducing random noise and preserving salient image details. Both synthetic and realistic experiments have demonstrated the superior performance of our DSPNet in terms of quantitative evaluations and visual image qualities.