³log 54 + ³log 18 – ³log 12=³log 54 + ³log 18 – ³log 12=

Let's simplify the expression \( \log_3 54 + \log_3 18 - \log_3 12 \).

Using the logarithm properties, we know that:

1. \( \log_b(x) + \log_b(y) = \log_b(x \cdot y) \)
2. \( \log_b(x) - \log_b(y) = \log_b\left(\frac{x}{y}\right) \)

So, we can start by combining the first two terms:

\[
\log_3 54 + \log_3 18 = \log_3 (54 \cdot 18)
\]

Now, calculate \( 54 \cdot 18 \):

\[
54 \times 18 = 972
\]

Thus, the expression becomes:

\[
\log_3 972 - \log_3 12
\]

Next, using the subtraction property:

\[
\log_3 972 - \log_3 12 = \log_3\left(\frac{972}{12}\right)
\]

Now, calculate \( \frac{972}{12} \):

\[
\frac{972}{12} = 81
\]

So, the expression simplifies to:

\[
\log_3 81
\]

Finally, recognize that \( 81 \) can be expressed as \( 3^4 \):

\[
\log_3 81 = \log_3 (3^4) = 4
\]

Thus, the value of the expression is:

\[
\boxed{4}
\]

Would you like further details on this, or do you have any questions?

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Learn how to simplify the logarithmic expression log 54 + log 18 - log 12 step-by-step using logarithm properties. This problem involves logarithmic calculations and is suitable for high school students.

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