O lvl A Maths: Indices and Logarithms

(i) By using the substitution y = 2^x, find the value of x such that

4^x+1 = 2 - 7(2^x).


(ii) Express 3^x(2^2x) = 7(5^x) in the form a^x = b. Hence find x.

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Answer:

(i) Since 4^x+1 = 4(4^x) = 4(2²)^x = 4(2^x)², the equation becomes

4(2^x)² = 2 - 7(2^x)

Substituting 2^x = y,
4y² = 2 - 7y
4y² +7y - 2 = -
(4y - 1)(y + 2) = 0
y = 0.25 or y = -2
2^x = 2^-2 or 2^x = -2 (N.A. as no solution)
Hence, x = -2


(ii) Since 3^x(2^2x) = 3^x(2²)^x = 12^x,
12^x = 7(5^x)
(12 / 5)^x = 7

Taking log on both sides:

x lg (12 / 5) = lg 7
x = lg 7 / lg (12/5)
x = 2.22

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