Using numerical analysis and the Power Rule of Logarithms, how can log(1343)

Question

log
⁡
(
1
343
)
be rewritten?(1 point)
Responses

3log−7
3
log
−
7
3 log minus 7

log−21
log
−
21
log minus 21

7log−3
7
log
−
3
7 log minus 3

−3log7

Answer 1

To rewrite \(\log(1343)\) and \(\log(1/343)\) using the properties of logarithms, we can apply the Power Rule of Logarithms.

The Power Rule states that \(\log(a^b) = b \cdot \log(a)\).

Let's analyze \(\log(1343)\) and \(\log(1/343)\):

1. Logarithm of \( 1/343 \): \[ \log\left(\frac{1}{343}\right) = \log(343^{-1}) = -\log(343) \]

2. Next, we note that \(343\) is \(7^3\): \[ \log(343) = \log(7^3) = 3 \cdot \log(7) \]

   So: \[ \log\left(\frac{1}{343}\right) = -\log(343) = -3 \cdot \log(7) \]

Now, we express everything together:

• \(\log(1343)\) is separate and is not transformed in this case as it is unrelated to the logarithm of \(343\).

From the responses given, the closest matching expression to \(-3 \cdot \log(7)\) is: \[ \mathbf{-3\log7} \]

Thus we can conclude: \[ \log(1343) + \log\left(\frac{1}{343}\right) = \log(1343) - 3 \log(7) \]

In the specific context of your question, we specifically needed to express \(\log(1/343)\), which we found to be: \[ \mathbf{-3\log7} \]