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

Answer 1

#x=2#

We would like to have an expression like

#log_4(a)=log_4(b)#, because if we had it, we could finish easily, observing that the equation would the solved if and only if #a=b#. So, let's do some manipulations:

The equation then rewrites as

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

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

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

Which is of course

#log_4(x/2)=log_4(x-1)#
Now we are in the desired form: since the logarithm is injective, if #log_4(a)=log_4(b)#, then necessarily #a=b#. In our case,
#log_4(x/2)=log_4(x-1) iff x/2 = x-1#
Which is easily solve into #x=2x-2#, which yields #x=2#
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Answer 2

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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Answer from HIX Tutor

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.

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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