Daniel is using logarithms to solve the equation 24x = 14. which of the following equations would be equivalent to his original expression? 2 log 4x = log 14 4x log 2 = log 14 2 log 4 = x log 14 x log 4 = 2 log 14

Just jumping in, without any preliminary simplification:  

log 24 + log x = log 14, so log x = (log 14) / (log 24).

Note that log 24= log 4*6 = log 4 + log 6.  This isn't very helpful because 6 does not show up in any of the four answer choices.

Because of this, I realilzed that you may have meant   24^x = 14, which is vastly different from 24x = 14.  It's essential that you use " ^ " to indicate exponentiation.

If 24^x = 14 is correct, then taking the log of both sides produces

x log 24 = log 14, and this is equivalent to x(log 2^3*3) = log 14.  Unfortunately, this doesn't match any of the 4 possible answers either (none of them contain "3").

Please, go back to the original question and ensure that you have copied it down correctly, as well as using " ^ " to indicate exponentiation, if applicable.

Yes the scientist can calculate for the distance between the Earth and the Sun by calculating the other angle and then using the sine formula.

SO the first step the scientist that have to do is to calculate for the other angle at the corner of the shooting star. Let’s call the third angle as angle z. We know that in a triangle, the total angle is 180 degrees, therefore:

z = 180 – x – 90

Now we have 2 angles and 1 side. We can now use the sine formula to solve for the distance of Earth and the sun. The sine formula is:

Side 1 / sin (angle opposite to Side 1) = Side 2 / sin (angle opposite to Side 2)

using that:

BC / sin (z) = y / sin (x)

Therefore BC is:

BC = y * sin (z) / sin (x)