Quadratic or logarithmic?

Algebra Level 3

The product of the two roots of 2014 x log 2014 x = x 2014 \sqrt{2014}x^{\log_{2014}{⁡x}} =x^{2014} is an integer. Find its units digit.


The answer is 6.

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1 solution

Prateek Saini
Jul 7, 2017

Let 201 4 x = a 2014^x=a . We get 201 4 a 2 + 0.5 = 201 4 2014 2014^{a^2+0.5}=2014^{2014} So a 2 2014 a + 0.5 = 0 a^2-2014a+0.5=0 If x 1 , x 2 x_1,x_2 were the solutions to the original equation and a 1 , a 2 a_1,a_2 for the second equation, x 1 × x 2 = 201 4 a 1 + a 1 = 201 4 2014 x_1\times x_2 = 2014^{a_1+a_1}=2014^{2014} The units digit of powers of 4 varies as 4 , 6 , 4 , 6... 4,6,4,6... and 2014 is even so the answer is 6

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