1 Answer

Neeraj Prasad Sharma · Answer 1 · 2023-07-16T17:17:53+0000

Answer:

x=2+2√( 26)

x=2-2√( 26)

Explanation:

Given logarithmic equation:

$\log_(10)(x^(2)-4x)=2$

$\textsf{Apply log law}: \quad \log_ab=c \iff a^c=b$

$\implies 10^2=x^2-4x$

$\implies 100=x^2-4x$

$\implies x^2-4x=100$

Add the square of half the coefficient of the term in x to both sides of the equation:

$\implies x^2-4x+\left((-4)/(2)\right)^2=100+\left((-4)/(2)\right)^2$

$\implies x^2-4x+\left(-2\right)^2=100+\left(-2}\right)^2$

$\implies x^2-4x+4=100+4$

$\implies x^2-4x+4=104$

Factor the perfect square trinomial on the left side of the equation:

$\implies (x-2)^2=104$

Square root both sides:

$\implies x-2=\pm √(104)$

$\implies x-2=\pm √(4 \cdot 26)$

$\implies x-2=\pm √(4) √( 26)$

$\implies x-2=\pm 2√( 26)$

Add 2 to both sides:

$\implies x=2\pm 2√( 26)$

Therefore, the solutions are:

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