Algebra Frenzy - Part 6

Algebra Level pending

Simplify 2 lg ⁡ 2 2 + lg ⁡ 2 lg ⁡ 5 + lg ⁡ 2 2 − lg2 + 1 \displaystyle 2 \lg^{2}\sqrt{2} + \lg\sqrt{2}\lg5+ \sqrt{\lg^{2}\sqrt{2}-\text{lg2}+1}


The answer is 1.

Alice Smith by Alice Smith , 20, China

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1 solution

Chris Lewis
May 20, 2019

Put x = lg ⁡ 2 x=\lg \sqrt2 . Note that x < 1 x<1 .

Then we have the relations lg ⁡ 2 = 2 x \lg 2 = 2x and lg ⁡ 5 = 1 − 2 x \lg 5 = 1-2x , and the expression becomes

2 x 2 + x ( 1 − 2 x ) + x 2 − 2 x + 1 2x^2+x(1-2x)+\sqrt{x^2-2x+1}

Since x < 1 x<1 , this is

2 x 2 + x ( 1 − 2 x ) + 1 − x = 1 2x^2+x(1-2x)+1-x=\boxed1

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