Given that 39^0.66 = x, 39^0.23 = y and x^z = y^2, then the value of z will be closest to which of the following?
0.69
Substitute x = 39^0.66 and y = 39^0.23 into x^z = y^2. This gives (39^0.66)^z = (39^0.23)^2, i.e. 39^(0.66z) = 39^(0.46).
Since the bases are equal and positive (and not 1), the exponents must be equal: 0.66z = 0.46.
Therefore z = 0.46 / 0.66 = 46/66 = 23/33 ≈ 0.6969…, which is closest to 0.69.
Hence, the correct answer is 0.69.
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