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Find the value of x in the equation:

2 * Log₂(x) + Log₄(x) - Log₈(x) = 0

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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.

User Shawnic Hedgehog
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