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Algebra Level 3

x log ⁡ 10 x = 1 0 2 − 3 log ⁡ 10 x + 2 ( log ⁡ 10 x ) 2 \large x^{\log_{10} x} = 10^{2-3\log_{10}x +2(\log_{10}x)^{2}}

Find the sum of all the solutions to the equation above.

111 10 100 110

Ir J by IR J , 21, Philippines

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1 solution

Chew-Seong Cheong
Sep 19, 2018

x log ⁡ 10 x = 1 0 2 − 3 log ⁡ 10 x + 2 log ⁡ 10 2 x Taking log ⁡ 10 on both sides log ⁡ 10 2 x = 2 − 3 log ⁡ 10 x + 2 log ⁡ 10 2 x \begin{aligned} x^{\log_{10} x} & = 10^{2-3\log_{10} x + 2\log^2_{10}x} & \small \color{#3D99F6} \text{Taking }\log_{10} \text{ on both sides} \\ \log^2_{10} x & = 2-3\log_{10} x + 2\log^2_{10}x \end{aligned}

⟹ log ⁡ 10 2 x − 3 log ⁡ 10 x + 2 log ⁡ 10 2 x + 2 = 0 ( log ⁡ 10 x − 1 ) ( log ⁡ 10 x − 2 ) = 0 \begin{aligned} \implies \log^2_{10} x -3\log_{10} x + 2\log^2_{10}x + 2 & = 0 \\ \left(\log_{10} x - 1\right) \left(\log_{10} x - 2\right) & = 0 \end{aligned}

⟹ { log ⁡ 10 x = 1 ⟹ x = 10 log ⁡ 10 x = 2 ⟹ x = 100 \implies \begin{cases} \log_{10} x = 1 & \implies x = 10 \\ \log_{10} x = 2 & \implies x = 100 \end{cases}

Therefore, the sum of solutions is 10 + 100 = 110 10+100 = \boxed{110} .

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Can you edit your LaTeX(The 5th line)?

X X - 2 years, 8 months ago

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Done. Thanks.

Chew-Seong Cheong - 2 years, 8 months ago

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