f.4 logarithmic functions

31. (2-log8-3log5)/(2log5-log0.25)
= (2-3log2-3log5)/(2log5+2log2)
= [2-3(log10)]/2(log10)
= (2-3)/2
= -1/2
34.given that log9=x,express log 30 in terms of x
log9=x
2log3=x
log30 = log3 + log10
= x/2 + 1
35.given that log3=x and log75=y,express log 5 in terms of x and y.
log3=x
log75=y
2log5 + log3 = y
2log5 + x = y
2log5 = y-x
log5 = (y-x)/2
12.3logx-4logy=0 (express x in terms of y)
logx^3 = log y^4
x^3 = y^4
x = y^4/3
25.log(x-5y)=2 --------(1)
logx-logy=1 -------(2)
From (1), x-5y = 100 -----(3)
From (2), log(x/y) = 1
x/y = 10
x = 10y ---------(4)
(3)-(4), -5y = 100 - 10y
5y = 100
y = 20
Sub y = 20 into (4)
x = 10 * 20
= 200
26.logy=log(2x)-1 --------(1)
logx=2log(y+1)^(1/2) ----------(2)
From (1), logy = log (2x/10)
y= 2x/10
10y = 2x ---------(3)
From (2), logx = log(y+1)
x = y+1 --------(4)
(3)-(4), -1 = 2x/10 - x
-1 = -8x/10
x = 10/8
x = 5/4
Sub x = 5/4 into (4)
5/4 = y + 1
y = 1/4