To simplify the expression:

\[
^3\log 54 + ^3\log 18 - ^3\log 12
\]

We use the logarithmic properties:
1. **Logarithm of a product**: \( ^a\log(xy) = ^a\log x + ^a\log y \)
2. **Logarithm of a quotient**: \( ^a\log\left(\frac{x}{y}\right) = ^a\log x - ^a\log y \)

Let’s simplify step by step.

1. Combine the first two logarithms using the product rule:
   \[
   ^3\log 54 + ^3\log 18 = ^3\log(54 \times 18) = ^3\log 972
   \]

2. Now subtract \( ^3\log 12 \):
   \[
   ^3\log 972 - ^3\log 12 = ^3\log\left(\frac{972}{12}\right) = ^3\log 81
   \]

3. Since \( 81 = 3^4 \), the logarithm becomes:
   \[
   ^3\log 81 = ^3\log 3^4 = 4
   \]

Thus, the value of the given expression is:

\[
4
\]

Would you like more details on any specific step?

Here are some related questions:
1. How does the change of base formula apply to logarithms?
2. Can you explain other properties of logarithms, like the power rule?
3. How would the result change if the logarithm base was different?
4. What are the common applications of logarithms in real-life problems?
5. How can we express logarithms in terms of natural logs?

**Tip**: Always remember to simplify logarithmic expressions step by step using product and quotient rules first!

Discover how to simplify the logarithmic expression ³log54 + ³log18 - ³log12 step-by-step using logarithmic product and quotient rules. Learn how to simplify complex logarithmic expressions, suitable for students in Grades 10-12.

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