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Answer:

To find the sum of all the solutions to the equation 2logx - log (2x-75) = 2, we can use logarithmic properties and algebraic manipulation.

Let's start by simplifying the equation:

2logx - log (2x-75) = 2

Using the properties of logarithms, we can rewrite it as:

log(x^2) - log (2x-75) = 2

Applying the quotient rule of logarithms, we have:

log(x^2 / (2x-75)) = 2

Converting the equation back to exponential form, we get:

x^2 / (2x-75) = 10^2

x^2 = 100(2x-75)

x^2 = 200x - 7500

Rearranging the equation, we have:

x^2 - 200x + 7500 = 0

Now we can solve this quadratic equation to find the solutions for x.

Using the quadratic formula:

x = (-b ± sqrt(b^2 - 4ac)) / (2a)

For our equation, a = 1, b = -200, and c = 7500.

Plugging in these values into the quadratic formula, we get:

x = (-(-200) ± sqrt((-200)^2 - 4(1)(7500))) / (2(1))

Simplifying further:

x = (200 ± sqrt(40000 - 30000)) / 2

x = (200 ± sqrt(10000)) / 2

x = (200 ± 100) / 2

x = 150 or x = 50

Now we can calculate the sum of these solutions:

Sum = 150 + 50 = 200

Therefore, the sum of all the solutions to the equation is 200, which corresponds to option (d).

EXPLANATION.

The sum of all the solutions to the equation.

⇒ 2㏒₁₀(x) - ㏒₁₀(2x - 75) = 2.

As we know that,

Properties of logarithms.

Let M and N arbitrary positive number such that a > 0, a ≠ 1, b > 0, b ≠ 1 then.

(1) ㏒ₐ(M/N) = ㏒ₐM - ㏒ₐN.

(2) ㏒ₐN^{α} = α㏒ₐN (α any real numbers).

Using this properties in this question, we get.

⇒ ㏒₁₀(x)² - ㏒₁₀(2x - 75) = 2.

⇒ ㏒₁₀[(x²)/(2x - 75)] = 2.

⇒ (x²)/(2x - 75) = 10².

⇒ (x²)/(2x - 75) = 100.

⇒ x² = 100(2x - 75).

⇒ x² = 200x - 7500.

⇒ x² - 200x + 7500 = 0.

Factorizes the equation into middle term splits, we get.

⇒ x² - 150x - 50x + 7500 = 0.

⇒ x(x - 150) - 50(x - 150) = 0.

⇒ (x - 50)(x - 150) = 0.

⇒ x = 50  and  x = 150.

∴ The sum of all the solutions of the equation is 150 + 50 = 200.