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.