If 2^x = 54^y = 6. calculate 2/x+1/y.

Level: Senior

Solution 1.

From the given equations,
log(2^x) = log(54^y) = log(6),
x*log(2) = y*log(54) = log(6),
x = log(6)/log(2) and y = log(6)/log(54).
Therefore
2/x+1/y
= 2*log(2)/log(6)+log(54)/log(6)
= [log(4)+log(54)]/log(6)
= log(4*54)/log(6)
= log(216)/log(6)
= log(6^3)/log(6)
= 3*log(6)/log(6)
= 3.

Solution 2.

Let a = log(2) and b = log(3). Since
2^x = 6,
so
x*log(2) = log(6) = log(2*3) = log(2)+log(3),
ax = a+b,
x = (a+b)/a.
Since
54^y = 6,
y*log(54) = log(6),
y*log(2*3^3) = log(2)+log(3),
y(a+3b) = a+b,
y = (a+b)/(a+3b).
Hence
2/x+1/y
= 2a/(a+b)+(a+3b)/(a+b)
= (3a+3b)/(a+b)
= 3.