SOLUTION: Having some serious problems with these 3 logs: 1.) 2(log 2x-log y)-(log 3 + 2 log 5) 2.) 350/(1+e to the negative x) = 300 : solve for x (not sure where to start) 3

Question 167993: Having some serious problems with these 3 logs:
1.) 2(log 2x-log y)-(log 3 + 2 log 5)
2.) 350/(1+e to the negative x) = 300 : solve for x (not sure where to start)
3.) "the fifth" log of (x+4) - "the fifth" log of x = 3 (again solving for x, but not sure where to start with this one...it just isn't working out)
Thanks for the help,
Matt G.
Answer by nerdybill(7384) (Show Source): You can put this solution on YOUR website!
1.)
2(log 2x-log y)-(log 3 + 2 log 5)
2(log 2x-log y)-(log 3 + log 5^2)
2(log 2x-log y)-(log 3 + log 25)
2(log 2x/y)-(log 3*25)
log (2x/y)^2 - log 75
log (4x^2/y^2) - log 75
log (4x^2/y^2)/75
log (4x^2/75y^2)
2.) 350/(1+e to the negative x) = 300 : solve for x (not sure where to start)
350/(1+e^(-x)) = 300
350 = 300(1+e^(-x))
350 = 300 + 300e^(-x)
50 = 300e^(-x)
50/300 = e^(-x)
5/30 = e^(-x)
1/6 = e^(-x)
ln(1/6) = -x
-1.7918 = -x
1.7918 = x

3.) "the fifth" log of (x+4) - "the fifth" log of x = 3
log5(x+4) - log5(x) = 3
log5[(x+4)/x] = 3
(x+4)/x = 5^3
(x+4)/x = 125
x+4 = 125x
4 = 124x
0.032 = x

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