Logarithm basics

Algebra Level 4

Find $\large\displaystyle\sum_{k=1}^{n} \left (x_{k} \right )$

where : $n$ = number of values of $x$ in equation :

$x^{[(\frac{3}{4})(\log_{2}x)^{2}+\log_{2}x-\frac{5}{4}]} = \sqrt{2}$ .

If your answer come as $\frac{ a+b\sqrt[c]{d}}{d}$ submit it as $a+b+c+d$ .

The answer is 18.

1 solution

Tom Engelsman
Mar 31, 2018

Let $x = 2^k, k \in \mathbb{R}$ , which after substitution into the above exponential equation yields:

$(2^k)^{\frac{3}{4} k^2 + k - \frac{5}{4}} = 2^{1/2};$

or $\frac{3}{4}k^3 + k^2 - \frac{5}{4}k - \frac{1}{2} = 0$

which factors into $(k+2)(k-1)(3k+1) = 0 \Rightarrow k = -2, 1, -\frac{1}{3}$ .

Hence, our required summation computes to:

$2^{-2} + 2^{-\frac{1}{3}} + 2^{1} = \frac{1}{4} + \frac{1}{2^{\frac{1}{3}}} + 2 = \frac{9}{4} + \frac{2^{\frac{2}{3}}}{2} = \frac{9 + 2^{5/3}}{4} = \boxed{\frac{9 + 2 \cdot 4^{1/3}}{4}}.$

Logarithm basics

The answer is 18.

1 solution

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