diff --git a/pytensor/tensor/rewriting/math.py b/pytensor/tensor/rewriting/math.py
index acc8becbfb..327e14951e 100644
--- a/pytensor/tensor/rewriting/math.py
+++ b/pytensor/tensor/rewriting/math.py
@@ -552,6 +552,29 @@ def local_sqrt_sqr(fgraph, node):
         return [new_out]
 
 
+@register_specialize
+@node_rewriter([log])
+def local_log_sqrt(fgraph, node):
+    x = node.inputs[0]
+
+    if (
+        not x.owner
+        or not isinstance(x.owner.op, Elemwise)
+        or not isinstance(x.owner.op.scalar_op, ps.Sqrt)
+    ):
+        return
+
+    # Case for log(sqrt(x)) -> 0.5 * log(x)
+    x = x.owner.inputs[0]
+    old_out = node.outputs[0]
+    new_out = mul(as_tensor_variable(0.5, dtype=x.dtype), log(x))
+    if new_out.dtype != old_out.dtype:
+        new_out = cast(new_out, old_out.dtype)
+
+    copy_stack_trace(node.out, new_out)
+    return [new_out]
+
+
 @register_specialize
 @node_rewriter([exp, expm1])
 def local_exp_log_nan_switch(fgraph, node):
diff --git a/tests/tensor/rewriting/test_math.py b/tests/tensor/rewriting/test_math.py
index a6e734ae82..43f65c2282 100644
--- a/tests/tensor/rewriting/test_math.py
+++ b/tests/tensor/rewriting/test_math.py
@@ -1989,6 +1989,18 @@ def test_exp_log_nested(self, nested_expression, expected_switches):
         assert len(ops_graph) == expected_switches
 
 
+def test_log_sqrt() -> None:
+    x = pt.tensor("x", shape=(None, None))
+    out = log(sqrt(x))
+
+    out = rewrite_graph(out, include=["specialize"])
+
+    assert utt.assert_equal_computations(
+        [out],
+        [mul(pt.as_tensor_variable([[0.5]], dtype=x.dtype), log(x))],
+    )
+
+
 class TestSqrSqrt:
     def setup_method(self):
         mode = get_default_mode()