Breast cancer is a serious public health problem and is one of the leading causes of cancer-related deaths in women worldwide. Early detection of the disease can significantly increase the chances of survival. However, manual analysis of mammogram mastery images is complex and time-consuming, which can lead to disagreements among experts. For this reason, automated diagnostic systems can play a significant role in increasing the accuracy and efficiency of diagnosis. In this study, we present an effective deep learning (DL) method, which classifies mammogram mastery images into cancer and noncancer categories using a collected dataset. Our model is pretrained based on the Inception V3 architecture. First, we run 5-fold cross-validation tests on the fully trained and fine-tuned Inception V3 model. Next, we apply a combined method based on likelihood and mean, where the fine-tuned Inception V3 model demonstrated superior performance in classification. Our DL model achieved 99% accuracy and 99% F1 score. In addition, interpretable AI techniques were used to enhance the transparency of the classification process. The finely tuned Inception V3 model demonstrated the highest performance in classification, confirming its effectiveness in automatic breast cancer detection. The experimental results clearly indicate that our proposed DL-based method for breast cancer image classification is highly effective, especially its application in image-based diagnostic methods. This study brings to the fore the huge potential of AI-based solutions, which can play a significant role in increasing the accuracy and reliability of breast cancer diagnosis.