## Screenshot: Code Snippet in Theorem Prover
### Overview
The image shows a code snippet from a theorem prover (likely Lean or similar), displayed in a dark-themed code editor. The text is color-coded, with syntax highlighting for keywords, variables, and operations. The top-left corner contains three circular buttons (red, yellow, green), typical of macOS window controls.
### Components/Axes
- **Top-left corner**: Three circular buttons (red, yellow, green) for window management.
- **Code content**:
- **Line 1**: `theorem mathd_algebra_484 : Real.log 27 / Real.log 3 = 3 := by` (yellow text).
- **Line 2**: `field_simp` (yellow text).
- **Line 3**: `rw [← Real.log_rpow]` (purple text).
- **Line 4**: `all_goals norm_num` (yellow text).
### Detailed Analysis
- **Line 1**: Declares a theorem named `mathd_algebra_484`, stating that `Real.log 27 / Real.log 3 = 3`. The `:= by` indicates the start of a proof.
- **Line 2**: `field_simp` is a tactic to simplify expressions using field axioms.
- **Line 3**: `rw [← Real.log_rpow]` applies a rewrite rule (`←` denotes leftward application) to `Real.log_rpow`, likely simplifying logarithmic expressions.
- **Line 4**: `all_goals norm_num` normalizes all goals in the proof, possibly converting them to numerical form.
### Key Observations
- The theorem proves that `log₃(27) = 3`, leveraging logarithmic identities (`log_b(a^c) = c log_b(a)`).
- The use of `field_simp` and `rw` suggests the proof relies on algebraic simplification and rewrite rules.
- `all_goals norm_num` implies the proof system automatically handles numerical normalization across all subgoals.
### Interpretation
This code snippet demonstrates a formal proof in a theorem prover, verifying the logarithmic identity `log₃(27) = 3`. The `field_simp` and `rw` tactics automate algebraic manipulations, while `all_goals norm_num` ensures numerical consistency. The color coding (yellow for declarations, purple for tactics) aids readability, reflecting the structured nature of formal proofs. The theorem’s validity hinges on the correctness of the underlying mathematical axioms and the prover’s rewrite rules.
