1(?) + 10(?) + 100(?)

Algebra Level 2

Find the sum of the solutions to the logarithmic equation x log ⁡ x = 1 0 2 − 3 log ⁡ x + 2 ( log ⁡ x ) 2 \large x^{\log x} = 10^{2 - 3 \log x + 2 (\log x )^{2}} where log ⁡ x \log x is the logarithm of x x to the base 10 10 .

110 10 100 111

Benedict Dimacutac by Benedict Dimacutac , 21, Philippines

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1 solution

Md Mehedi Hasan
Nov 2, 2017

x log ⁡ x = 1 0 2 − 3 log ⁡ x + 2 ( log ⁡ x ) 2 ⇒ l o g x l o g x = 2 − 3 l o g x + 2 ( l o g x ) 2 ⇒ l o g x × l o g x = 2 − 3 l o g x + 2 ( l o g x ) 2 ⇒ ( l o g x ) 2 = 2 − 3 l o g x + 2 ( l o g x ) 2 ⇒ ( l o g x ) 2 − 3 l o g x + 2 = 0 ⇒ y 2 − 3 y + 2 = 0 Let, log x=y ⇒ ( y − 2 ) ( y − 1 ) = 0 ⇒ y = 2 o r , y = 1 ⇒ l o g x = 2 ⇒ l o g x = 1 ∴ x = 100 ∴ x = 10 \quad \LARGE {\quad x^{\log x} = 10^{2 - 3 \log x + 2 (\log x )^{2}} \\ \Rightarrow log x^{log x}=2-3log x+2(log x)^2 \\ \Rightarrow log x\times log x=2-3log x+2(log x)^2 \\ \Rightarrow (log x)^2=2-3log x+2(log x)^2\\ \Rightarrow (log x)^2-3log x+2=0\\ \Rightarrow y^2-3y+2=0 \quad \boxed{\color{#3D99F6}\text{Let, log x=y}}\\ \Rightarrow (y-2)(y-1)=0\\ \Rightarrow y=2\quad \quad \quad \quad \quad or, y=1\\ \Rightarrow log x=2\quad \quad \Rightarrow log x=1\\ \therefore x=100\quad \quad \quad\therefore x=10}

Now the summation of solution is = 100 + 10 = 110 =100+10=\boxed{110}

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excellently solved!

Mohammad Khaza - 3 years, 7 months ago

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