log 3^54+log 3^45-log3^30

Answer 1

To simplify the expression \( \log 3^{54} + \log 3^{45} - \log 3^{30} \), we can use the properties of logarithms:

1. The property \( \log a^b = b \log a \)
2. The property \( \log a + \log b = \log (ab) \)
3. The property \( \log a - \log b = \log \left(\frac{a}{b}\right) \)

First, apply the first property to each logarithmic term:

\[ \log 3^{54} = 54 \log 3 \] \[ \log 3^{45} = 45 \log 3 \] \[ \log 3^{30} = 30 \log 3 \]

Now substitute these back into the original expression:

\[ 54 \log 3 + 45 \log 3 - 30 \log 3 \]

Next, combine the terms:

\[ (54 + 45 - 30) \log 3 = 69 \log 3 \]

Thus, the simplified result is:

\[ \log 3^{69} \]

Finally, we can express this as:

\[ \log 3^{69} \] or simply \( 69 \log 3 \).