Find the number of values of x for which 8x+27x12x+18x=76

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Question

Find the number of values of x for which

\frac{8^{x} + 27^{x}}{12^{x} + 18^{x}} = \frac{7}{6}

Answer: Option B
:
B
Let

2^{x} = a

and

3^{x} = b,

the equation becomes

\frac{a^{3} + b^{3}}{a^{2 b} + b^{2 a}} = \frac{7}{6}

o r \frac{a^{2} - a b + b^{2}}{a b} = \frac{7}{6}

o r 6 a^{2} - 13 a b + 6 b^{2} = 0

(2 a - 3 b) (3 a - 2 b) = 0

Therefore

2^{x + 1} = 3^{x + 1}

or

2^{x - 1} = 3^{x - 1}

which implies that x= -1 and x= 1. Hence only two solutions exist.

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