Question 1012454
you were on the right track.  log 4 (2)=1/2, and dividing by 1/2 is multiplying by 2, which is what I did when I converted to base 2, rather than base 4.

4log4((x+1)^(1/2)) =log2(5-7x)
log 4 (x+1)^2=log 2 (5-7x)
log2 (x+1)^2/log2 (4)=log 2 (5-7x)
log2(4)=2, multiply by 2 both sides
log 2(x+1)^2=log 2 (5-7x)^2
(x+1)^2=(5-7x)^2
x^2+2x+1=25-70x+49x^2
0=24-72x+48x^2
divide both sides by 24
2x^2-3x+1=0
(2x-1)(x-1)=0
x=1/2, 1
check back
4 log 4(2^(1/2)-log 2 (5-7)=0; the second term does not exist.
4 log 4 (3/2)^(1/2)-log2 (5-3.5)=0
log 4 (3/2)^2=log 2 (3/2)
log 2 (3/2)^2/log 2 (4)=log 2 (3/2)
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The second term is 0.585 or 2^0.585=1.5
the first term is 4 log 4(3/2)^(1/2)
3/2^(1/2)=1.225
4 log 4 (1.225), and log 4(1.225)=0.1465
0.1465*4=0.586, rounding error.
log 2 (4)=2
multiply through by 2
log 2 (3/2)^2=2 log 2 (3/2)=log 2 (3/2)^2
x=1/2