SOLUTION: If x=2 to the power y write 2 to the power 2y and 2 to the 2y+1 in terms of x and solve the equation 2 to the 2y+1 -5(2 to the y)+2=0

Question 366565: If x=2 to the power y write 2 to the power 2y and 2 to the 2y+1 in terms of x and
solve the equation 2 to the 2y+1 -5(2 to the y)+2=0
Found 2 solutions by amoresroy, CharlesG2:
Answer by amoresroy(361) (Show Source): You can put this solution on YOUR website!
If x=2 to the power y write 2 to the power 2y and 2 to the 2y+1 in terms of x and
x = 2^y
x^2 = 2^2y
2x^2 = 2^(2y+1)
solve the equation 2 to the 2y+1 -5(2 to the y)+2=0
Express above euation in terms of x
2x^2 -5x +2 = 0
Solve for x by factoring method
(2x-1)(x-2) = 0
When 2x-1 = 0
2x = 1
x =1/2
substitute x = 1/2 in equation x = 2^y
2^y = 1/2
Get log of both sides
log 2^y = log 1/2
y log 2 = log 1/2
y = [log 1/2]/log 2
When x-2 = 0
x = 2
substitute x = 2 in equation x = 2^y
2^y = 2
Get log of both sides
log 2^y = log 2
y log 2 = log 2
y = 1
Answer by CharlesG2(834) (Show Source): You can put this solution on YOUR website!
If x=2 to the power y write 2 to the power 2y and 2 to the 2y+1 in terms of x and solve the equation 2 to the 2y+1 -5(2 to the y)+2=0

x = 2^y
2^(2y) is what in terms of x
2^(2y + 1) is what in terms of x
solve 2^(2y + 1) - 5(2^y) + 2 = 0

logarithmic rule: if b^y = x then logb x = y
in this case b = 2

if x = 2^y then log2 x = y

2^(2y) = 2^(2log2 x) = 2^(log2 x^2)
logarithmic rule: logb (m^n) = nlogb m

2^(2y + 1) = 2^(log2 x^2 + 1)
1 = log2 (2)
2^(2y + 1) = 2^(log2 x^2 + log2 2)
logarithmic rule: logb m + logb n = logb mn
2^(2y + 1) = 2^(log2 x^2 + log2 2) = 2^(log2 2x^2)

solve 2^(2y + 1) - 5(2^y) + 2 = 0
2^(log2 2x^2) - 5x + 2 = 0
2^(log2 2x^2) = 5x - 2
log2 (5x - 2) = log2 (2x^2)
if these 2 logs equal then 5x - 2 = 2x^2
-2x^2 + 5x - 2 = 0
2x^2 - 5x + 2 = 0
x^2 - (5/2)x + 1 = 0
(x - 2)(x - (1/2)) = 0, check with FOIL, First Outer Inner Last
x^2 - (1/2)x - 2x + 1 --> x^2 - (1/2)x - (4/2)x + 1
x^2 - (1/2)x - (4/2)x + 1 = x^2 - (5/2)x + 1, yes
x = 2 or x = 1/2
x = 2^y --> log2 x = y
log2 2 = 1
log2 (1/2) = -1

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