diff --git a/pymc/logprob/transforms.py b/pymc/logprob/transforms.py
index 82aac0fa88..46b3a163c4 100644
--- a/pymc/logprob/transforms.py
+++ b/pymc/logprob/transforms.py
@@ -374,12 +374,15 @@ def measurable_transform_logprob(op: MeasurableTransform, values, *inputs, **kwa
     else:
         input_logprob = logprob(measurable_input, backward_value)
 
-    if input_logprob.ndim < value.ndim:
-        # Do we just need to sum the jacobian terms across the support dims?
-        raise NotImplementedError("Transform of multivariate RVs not implemented")
-
     jacobian = op.transform_elemwise.log_jac_det(value, *other_inputs)
 
+    if input_logprob.ndim < value.ndim:
+        # For multivariate variables, the Jacobian is diagonal.
+        # We can get the right result by summing the last dimensions
+        # of `transform_elemwise.log_jac_det`
+        ndim_supp = value.ndim - input_logprob.ndim
+        jacobian = jacobian.sum(axis=tuple(range(-ndim_supp, 0)))
+
     # The jacobian is used to ensure a value in the supported domain was provided
     return pt.switch(pt.isnan(jacobian), -np.inf, input_logprob + jacobian)
 
@@ -674,7 +677,7 @@ def backward(self, value, *inputs):
 
     def log_jac_det(self, value, *inputs):
         scale = self.transform_args_fn(*inputs)
-        return -pt.log(pt.abs(scale))
+        return -pt.log(pt.abs(pt.broadcast_to(scale, value.shape)))
 
 
 class LogTransform(RVTransform):
@@ -892,7 +895,8 @@ def log_jac_det(self, value, *inputs):
         det = 0.0
         for det_ in det_list:
             if det_.ndim > ndim0:
-                det += det_.sum(axis=-1)
+                ndim_diff = det_.ndim - ndim0
+                det += det_.sum(axis=tuple(range(-ndim_diff, 0)))
             else:
                 det += det_
         return det
diff --git a/tests/logprob/test_transforms.py b/tests/logprob/test_transforms.py
index 29c8dc0ea7..5e224afabe 100644
--- a/tests/logprob/test_transforms.py
+++ b/tests/logprob/test_transforms.py
@@ -626,8 +626,8 @@ def test_chained_transform():
 
     log_jac_det = ch.log_jac_det(x_val_forward, *x.owner.inputs, scale, loc)
     assert np.isclose(
-        log_jac_det.eval(),
-        -np.log(scale) - np.sum(np.log(x_val_forward - loc)),
+        pt.sum(log_jac_det).eval(),
+        np.sum(-np.log(scale) - np.log(x_val_forward - loc)),
     )
 
 
@@ -964,3 +964,28 @@ def scan_step(prev_innov):
         "innov": np.full((4,), -0.5),
     }
     np.testing.assert_allclose(logp_fn(**test_point), ref_logp_fn(**test_point))
+
+
+@pytest.mark.parametrize("shift", [1.5, np.array([-0.5, 1, 0.3])])
+@pytest.mark.parametrize("scale", [2.0, np.array([1.5, 3.3, 1.0])])
+def test_multivariate_transform(shift, scale):
+    mu = np.array([0, 0.9, -2.1])
+    cov = np.array([[1, 0, 0.9], [0, 1, 0], [0.9, 0, 1]])
+    x_rv_raw = pt.random.multivariate_normal(mu, cov=cov)
+    x_rv = shift + x_rv_raw * scale
+    x_rv.name = "x"
+
+    x_vv = x_rv.clone()
+    logp = factorized_joint_logprob({x_rv: x_vv})[x_vv]
+    assert_no_rvs(logp)
+
+    x_vv_test = np.array([5.0, 4.9, -6.3])
+    scale_mat = scale * np.eye(x_vv_test.shape[0])
+    np.testing.assert_almost_equal(
+        logp.eval({x_vv: x_vv_test}),
+        sp.stats.multivariate_normal.logpdf(
+            x_vv_test,
+            shift + mu * scale,
+            scale_mat @ cov @ scale_mat.T,
+        ),
+    )