Final answer:
Using the properties of logarithms, the given logarithmic equation simplifies to a single base 2 logarithm of x equating to zero, which means x = 1.
Step-by-step explanation:
To find the value of x in the given logarithmic equation, we need to understand and apply the properties of logarithms. We have the equation:
2 * Log2(x) + Log4(x) - Log8(x) = 0
To simplify, first recognize that Log4(x) and Log8(x) can be converted to the base 2 logarithms because 4 = 22 and 8 = 23. Applying the change of base property, we have:
2 * Log2(x) + (1/2)*Log2(x) - (1/3)*Log2(x) = 0
Combine the terms:
(2 + 1/2 - 1/3) * Log2(x) = 0
Simplify the coefficients:
(6/3 + 1/2 - 1/3) * Log2(x) = 0
(5/3) * Log2(x) = 0
To find x, divide both sides by (5/3):
Log2(x) = 0
The logarithm of a number to its base is zero means the number is 1:
x = 20 = 1
Therefore, the value of x that satisfies the equation is 1.