2×log8(2x)+log8(x-1)^2=4/3

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To solve this logarithmic equation, we first need to simplify the terms using the properties of logarithms.

Given equation: 2*log8(2x) + log8(x-1)^2 = 4/3

Using the properties of logarithms, we can rewrite log8(2x) as log8(2) + log8(x). Similarly, log8(x-1)^2 can be rewritten as 2*log8(x-1).

Therefore, the equation becomes: 2(log8(2) + log8(x)) + 2log8(x-1) = 4/3.

Now, we will use the property of logarithms which states that a*logb(c) = logb(c)^a to simplify the equation further.

The equation becomes: log8(2^2) + 2*log8(x) + log8((x-1)^2) = 4/3
Simplify this further:
log8(4) + log8(x^2) + log8((x-1)^2) = 4/3

Now, we can combine the logarithmic terms using the property of adding logarithms of the same base.

log8((4)(x^2)((x-1)^2)) = 4/3

Now we can express the equation using the formula logb(mn) = logb(m) + logb(n).

log8(4) + log8(x^2) + log8((x-1)^2) = 4/3
log8(4(x^2)((x-1)^2)) = 4/3
log8(4x^2(x-1)^2) = 4/3

Now rewrite the equation in exponential form:
8^(4/3) = 4x^2(x-1)^2

Simplify the left side:
(2^3)^4/3 = 4x^2(x-1)^2
2^4 = 4x^2(x-1)^2
16 = 4x^2(x-1)^2

Now, we can solve for x by dividing both sides by 4 and then taking the square root of both sides:
16/4 = x^2(x-1)^2
4 = x^2(x-1)^2
2 = x(x-1)

Now, we can solve for x by setting x(x-1) = 2 to zero:
x^2 - x - 2 = 0
(x - 2)(x + 1) = 0
x = 2 or x = -1

Therefore, the solutions to the given logarithmic equation are x = 2 and x = -1.