# What is x if #log_4(8x ) - 2 = log_4 (x-1)#?

We would like to have an expression like

The equation then rewrites as

But we're still not happy, because we have the difference of two logarithms in the left member, and we want a unique one. So we use

So, the equation becomes

Which is of course

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To find x, we first need to combine the logarithmic terms using logarithmic properties. Then, we solve for x:

log_4(8x) - 2 = log_4(x - 1)

Using the property: log_a(b) - log_a(c) = log_a(b/c):

log_4((8x) / 4^2) = log_4(x - 1)

log_4(8x / 16) = log_4(x - 1)

Simplify the expression inside the logarithm:

log_4(1/2) + log_4(x) = log_4(x - 1)

Now, using the property: log_a(b) + log_a(c) = log_a(b * c):

log_4(1/2 * x) = log_4(x - 1)

log_4(x/2) = log_4(x - 1)

Now, we equate the arguments:

x / 2 = x - 1

Solve for x:

x - x / 2 = 1

(2x - x) / 2 = 1

x / 2 = 1

x = 2 × 1

x = 2

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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