We can simplify the expression step by step using logarithmic properties. The expression is:

$4 \log 54 + 3 \log 18 - 3 \log 12$

Step 1: Apply the logarithmic power rule

The logarithmic power rule states that $a \log b = \log b^a$. Applying this to each term:

$4 \log 54 = \log 54^4$ $3 \log 18 = \log 18^3$ $3 \log 12 = \log 12^3$

So the expression becomes:

$\log 54^4 + \log 18^3 - \log 12^3$

Step 2: Use the logarithmic addition and subtraction rules

The logarithmic addition rule states that $\log a + \log b = \log (a \times b)$, and the subtraction rule states that $\log a - \log b = \log \left(\frac{a}{b}\right)$. Applying these rules:

$\log (54^4 \times 18^3) - \log 12^3 = \log \left( \frac{54^4 \times 18^3}{12^3} \right)$

Step 3: Simplify the expression

Now calculate the powers of the numbers:

• $54^4 = 84934656$
• $18^3 = 5832$
• $12^3 = 1728$

So the expression becomes:

$\log \left( \frac{84934656 \times 5832}{1728} \right)$

First, calculate $84934656 \times 5832 = 495683142912$.

Then, divide by 1728:

$\frac{495683142912}{1728} = 286824192$

Final expression:

$\log (286824192)$

So, the simplified form of the expression is:

$\log (286824192)$

Would you like more details, or do you have any questions?

Related questions:

1. What are the basic properties of logarithms?
2. How do you calculate large powers efficiently?
3. How do logarithmic addition and subtraction work?
4. Can logarithmic expressions always be simplified?
5. What are some real-life applications of logarithms?

Tip: Remember, logarithms are the inverse of exponents, so they help simplify expressions with large powers and multiplications.