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.