To solve the equation log(2x) - 5log(x) + 6 = 0 and find the value of x, we can use logarithmic properties and algebraic manipulation.

Let's go through each option to determine the correct answer:

Option A) e^2 If x = e^2, then 2x = 2e^2. Substituting these values into the equation, we get: log(2e^2) - 5log(e^2) + 6 = 0 2log(e) + 2log(2) - 5(2log(e)) + 6 = 0 2 + 2log(2) - 10 - 10log(e) + 6 = 0 -2 - 10log(e) + 6 + 2log(2) = 0 -2 - 10 + 6 + 2log(2) = 0 -6 + 2log(2) = 0 2log(2) = 6 log(2^2) = 6 4 = 6 (which is not true)

Option A is incorrect.

Option B) e^3 If x = e^3, then 2x = 2e^3. Substituting these values into the equation, we get: log(2e^3) - 5log(e^3) + 6 = 0 3log(e) + 2log(2) - 15log(e) + 6 = 0 3 + 2log(2) - 15 - 15log(e) + 6 = 0 -12 - 15log(e) + 2log(2) = 0 -12 - 15 + 2log(2) = 0 -27 + 2log(2) = 0 2log(2) = 27 log(2^2) = 27 4 = 27 (which is not true)

Option B is incorrect.

Option C) e^2 or e^3 If we substitute x = e^2, we get: log(2e^2) - 5log(e^2) + 6 = 0 2log(e) + 2log(2) - 10log(e) + 6 = 0 2 + 2log(2) - 10 - 10log(e) + 6 = 0 -8 - 10log(e) + 2log(2) = 0 -8 - 10 + 2log(2) = 0 -18 + 2log(2) = 0 2log(2) = 18 log(2^2) = 18 4 = 18 (which is not true)

If we substitute x = e^3, we get: log(2e^3) - 5log(e^3) + 6 = 0 3log(e) + 2log(2) - 15log(e) + 6 = 0 3 + 2log(2) - 15 - 15log(e) + 6 = 0 -12 - 15log(e) + 2log(2) = 0 -12 - 15 + 2log(2) = 0 -27 + 2log(2) = 0 2log(2) = 27 log(2^2) = 27 4 = 27 (which is not true)

Both x = e^2 and x = e^3 do not satisfy the equation.

Option C is incorrect.

Option D) None of these Since none of the options satisfy the equation, we can conclude that the correct answer is option D) None of these.

Therefore, the correct answer is D) None of these.

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• Wrong explanation

To solve this question, let's start by understanding the given equation:

a:b = c:d

This equation represents a proportion, where the ratio of a to b is equal to the ratio of c to d.

Now, let's consider the expression (pa+qc)/(pb+qd). We can simplify this expression by multiplying both the numerator and denominator by the same value.

Multiplying the numerator by p and the denominator by q, we get:

(pa+qc)/(pb+qd) = (p*a + q*c)/(p*b + q*d)

Now, let's compare this expression to the given equation a:b = c:d.

The numerator of the expression (pa+qc)/(pb+qd) is p*a + q*c, which has the same ratio as a and c.

The denominator of the expression (pa+qc)/(pb+qd) is p*b + q*d, which has the same ratio as b and d.

Therefore, the expression (pa+qc)/(pb+qd) has the same ratio as a:b and c:d.

Hence, the correct answer is option D) a:b and c:d.

• Thanks
• Wrong explanation