diff --git a/ch07/04_preference-tuning-with-dpo/dpo-from-scratch.ipynb b/ch07/04_preference-tuning-with-dpo/dpo-from-scratch.ipynb
index fe5a370..edfd9dc 100644
--- a/ch07/04_preference-tuning-with-dpo/dpo-from-scratch.ipynb
+++ b/ch07/04_preference-tuning-with-dpo/dpo-from-scratch.ipynb
@@ -1886,8 +1886,9 @@
     "reference_logratios = reference_chosen_logprobs - reference_rejected_logprobs\n",
     "```\n",
     "\n",
-    "- These lines above calculate the difference in log probabilities (logits) for the chosen and rejected samples for both the policy model and the reference model:\n",
-    "$\\log \\left( \\frac{\\pi_\\theta (y_w \\mid x)}{\\pi_\\theta (y_l \\mid x)} \\right) \\quad \\text{and} \\quad \\log \\left( \\frac{\\pi_{\\text{ref}}(y_w \\mid x)}{\\pi_{\\text{ref}}(y_l \\mid x)} \\right)$; this is due to $\\log\\left(\\frac{a}{b}\\right) = \\log a - \\log b$"
+    "- These lines above calculate the difference in log probabilities (logits) for the chosen and rejected samples for both the policy model and the reference model (this is due to $\\log\\left(\\frac{a}{b}\\right) = \\log a - \\log b$):\n",
+    "\n",
+    "$$\\log \\left( \\frac{\\pi_\\theta (y_w \\mid x)}{\\pi_\\theta (y_l \\mid x)} \\right) \\quad \\text{and} \\quad \\log \\left( \\frac{\\pi_{\\text{ref}}(y_w \\mid x)}{\\pi_{\\text{ref}}(y_l \\mid x)} \\right)$$"
    ]
   },
   {
@@ -1897,8 +1898,10 @@
     "id": "5458d217-e0ad-40a5-925c-507a8fcf5795"
    },
    "source": [
-    "- Next, the code `logits = model_logratios - reference_logratios` computes the difference between the model's log ratios and the reference model's log ratios, i.e., $\\beta \\log \\left( \\frac{\\pi_\\theta (y_w \\mid x)}{\\pi_{\\text{ref}} (y_w \\mid x)} \\right)\n",
-    "- \\beta \\log \\left( \\frac{\\pi_\\theta (y_l \\mid x)}{\\pi_{\\text{ref}} (y_l \\mid x)} \\right)$\n"
+    "- Next, the code `logits = model_logratios - reference_logratios` computes the difference between the model's log ratios and the reference model's log ratios, i.e., \n",
+    "\n",
+    "$$\\beta \\log \\left( \\frac{\\pi_\\theta (y_w \\mid x)}{\\pi_{\\text{ref}} (y_w \\mid x)} \\right)\n",
+    "- \\beta \\log \\left( \\frac{\\pi_\\theta (y_l \\mid x)}{\\pi_{\\text{ref}} (y_l \\mid x)} \\right)$$\n"
    ]
   },
   {
@@ -1908,8 +1911,10 @@
     "id": "f18e3e36-f5f1-407f-b662-4c20a0ac0354"
    },
    "source": [
-    "- Finally, `losses = -F.logsigmoid(beta * logits)`  calculates the loss using the log-sigmoid function; in the original equation, the term inside the expectation is $\\log \\sigma \\left( \\beta \\log \\left( \\frac{\\pi_\\theta (y_w \\mid x)}{\\pi_{\\text{ref}} (y_w \\mid x)} \\right)\n",
-    "- \\beta \\log \\left( \\frac{\\pi_\\theta (y_l \\mid x)}{\\pi_{\\text{ref}} (y_l \\mid x)} \\right) \\right)$"
+    "- Finally, `losses = -F.logsigmoid(beta * logits)`  calculates the loss using the log-sigmoid function; in the original equation, the term inside the expectation is \n",
+    "\n",
+    "$$\\log \\sigma \\left( \\beta \\log \\left( \\frac{\\pi_\\theta (y_w \\mid x)}{\\pi_{\\text{ref}} (y_w \\mid x)} \\right)\n",
+    "- \\beta \\log \\left( \\frac{\\pi_\\theta (y_l \\mid x)}{\\pi_{\\text{ref}} (y_l \\mid x)} \\right) \\right)$$"
    ]
   },
   {
@@ -3089,7 +3094,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.11.4"
+   "version": "3.10.6"
   }
  },
  "nbformat": 4,