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The given expression is:

\[
^3\log54 - ^3\log4 + ^3\log6
\]

Here, the notation "^3\log" refers to the logarithm with base 3.

Let's simplify this step by step using logarithmic properties:

### Step 1: Use the logarithmic subtraction rule
For logarithms with the same base, subtraction can be expressed as a division:
\[
^3\log54 - ^3\log4 = ^3\log\left(\frac{54}{4}\right) = ^3\log13.5
\]

### Step 2: Use the logarithmic addition rule
For logarithms with the same base, addition can be expressed as multiplication:
\[
^3\log13.5 + ^3\log6 = ^3\log\left(13.5 \times 6\right)
\]
\[
^3\log(13.5 \times 6) = ^3\log81
\]

### Step 3: Simplify the logarithmic value
Now, simplify \( ^3\log81 \). Since \( 81 = 3^4 \), the logarithm becomes:
\[
^3\log81 = ^3\log(3^4) = 4
\]

Thus, the simplified form of the expression is:
\[
4
\]

Let me know if you need more clarification or details!

### Relative Questions:
1. What are the general properties of logarithms that were used in this problem?
2. How can you convert between different logarithmic bases?
3. Why does \( ^3\log(3^4) \) simplify to 4?
4. What is the relationship between exponentiation and logarithms?
5. Can you find other examples where logarithmic subtraction and addition are useful?

### Tip:
Always remember that logarithmic operations such as addition and subtraction can simplify complex expressions significantly by using multiplication and division rules.

Learn how to simplify the expression ^3log54 - ^3log4 + ^3log6 using the properties of logarithms, including subtraction and addition rules. This step-by-step solution is suitable for high school students (Grades 10-12).

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