Logarithm

Algebra Level 2

Find the value of x x satisfying 2 log ⁡ x log ⁡ ( 5 x − 4 ) = 1 \dfrac{2 \log x}{\log (5x-4)} = 1 .

4 5 3 1 2

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Devansh Agarwal by DEVANSH AGARWAL , 20, India

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1 solution

Devansh Agarwal
May 9, 2016

2 log ⁡ x log ⁡ ( 5 x − 4 ) = 1 2 log ⁡ x = log ⁡ ( 5 x − 4 ) log ⁡ ( x 2 ) = log ⁡ ( 5 x − 4 ) x 2 = 5 x − 4 x 2 − 5 x + 4 = 0 ( x − 4 ) ( x − 1 ) = 0 ⟹ x = 1 , 4 \begin{aligned} \frac{2\log x}{\log(5x-4)} & = 1 \\ 2\log x&=\log(5x-4) \\ \log(x^2)&=\log(5x-4) \\ x^2&=5x-4 \\ x^2-5x+4&=0 \\ (x-4)(x-1)&=0 \\ \implies x & =1,4 \end{aligned}

But x x cannot be equal to 1 1 because, log ⁡ 1 = 0 \log 1=0 , therefore, x = 4 x=4 .

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Devanash, I have added the LaTex codes for you. You can see them when you click edit. You can also see the codes by placing your mouse cursor on the formulas. Hope that you can learn it up. You can also install daum equation editor for free.

Note also that x ≠ 1 x \ne 1 not because log ⁡ 1 = 0 \log 1 = 0 , but the LHS of the equation is not defined.

Chew-Seong Cheong - 5 years, 1 month ago

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Thank you very much

DEVANSH AGARWAL - 5 years ago

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