Prostatitis: Given the patient's symptoms of low back pain, mild prostatic tenderness without a mass, and leakage of semen, prostatitis is a likely diagnosis. The absence of dysuria, hematuria, and systemic symptoms such as fever also points towards a non-bacterial or chronic form of prostatitis.

Urinary Tract Infection (UTI): Although the patient denies dysuria and hematuria, a UTI could still be considered, especially if the symptoms are mild or atypical. The presence of some leakage of semen could be related to urethral irritation.
Chronic Pelvic Pain Syndrome: This condition is characterized by chronic pain in the pelvic region, which could explain the patient's low back pain and mild prostatic tenderness.
Benign Prostatic Hyperplasia (BPH): Given the family history of prostate trouble, BPH could be a consideration, although it is less common in a 34-year-old male.

Testicular Cancer: Although rare, testicular cancer can present with low back pain due to metastasis. The absence of a testicular mass on examination does not rule out this diagnosis.
Spinal Cord Compression: This is a medical emergency that can present with low back pain and urinary symptoms. Although the patient's symptoms are mild, it is essential to consider this diagnosis to avoid missing a potentially devastating condition.
Prostate Cancer: Although the patient's age and lack of systemic symptoms make this diagnosis less likely, a family history of prostate trouble increases the risk.

Reiter's Syndrome: A form of reactive arthritis that can present with urethritis, conjunctivitis, and arthritis. The patient's symptoms do not fully align with this diagnosis, but it could be considered if other diagnoses are ruled out.
Spondyloarthritis: A group of inflammatory diseases that can cause low back pain and urethritis. The patient's symptoms are not typical for this diagnosis, but it could be considered if other diagnoses are ruled out.