Answer: <em>3. The correct option is: A) 3.57 decibels.</em>
<em>6. The correct option is: B)
</em>
Step-by-step explanation:
3. Sound intensity is cut to 44% of its original level. That means, if the original sound intensity was 100, then now sound intensity will be 44.
So,
and ![I= 44](https://tex.z-dn.net/?f=I%3D%2044)
Using the formula
, we will get.....
![L= 10log(\frac{44}{100})\\ \\ L= 10log(0.44)\\ \\ L= 10(-0.3565...)=-3.565...\approx -3.57](https://tex.z-dn.net/?f=L%3D%2010log%28%5Cfrac%7B44%7D%7B100%7D%29%5C%5C%20%5C%5C%20L%3D%2010log%280.44%29%5C%5C%20%5C%5C%20L%3D%2010%28-0.3565...%29%3D-3.565...%5Capprox%20-3.57)
<em>(Rounded to the nearest hundredth)</em>
So, the loudness would be reduced by 3.57 decibels.
6. Given expression is: ![2ln(x) - 5ln(c)](https://tex.z-dn.net/?f=2ln%28x%29%20-%205ln%28c%29)
First applying the property
, we will get.....
![ln(x^2)-ln(c^5)](https://tex.z-dn.net/?f=ln%28x%5E2%29-ln%28c%5E5%29)
Now using the formula,
, we will get....
![ln(x^2)-ln(c^5) = ln(\frac{x^2}{c^5})](https://tex.z-dn.net/?f=ln%28x%5E2%29-ln%28c%5E5%29%20%3D%20ln%28%5Cfrac%7Bx%5E2%7D%7Bc%5E5%7D%29)
Thus, the answer as a single natural logarithm is ![ln(\frac{x^2}{c^5})](https://tex.z-dn.net/?f=ln%28%5Cfrac%7Bx%5E2%7D%7Bc%5E5%7D%29)