solve for x 16^x-1=1024^x-1

To solve the equation \( 16^{x-1} = 1024^{x-1} \), we first express both sides of the equation in terms of powers of 2.

We know:

Thus, we can rewrite the equation as:

\[ (2^4)^{x-1} = (2^{10})^{x-1} \]

This simplifies to:

\[ 2^{4(x-1)} = 2^{10(x-1)} \]

Since the bases are the same, we can set the exponents equal to each other:

\[ 4(x-1) = 10(x-1) \]

Now we can distribute on both sides:

\[ 4x - 4 = 10x - 10 \]

Next, we rearrange the equation to isolate \( x \):

\[ -4 + 10 = 10x - 4x \] \[ 6 = 6x \]

Now we solve for \( x \):

\[ x = \frac{6}{6} = 1 \]

To confirm, we can substitute \( x = 1 \) back into the original equation:

\[ 16^{1-1} = 1024^{1-1} \] \[ 16^0 = 1024^0 \] \[ 1 = 1 \]

This is true, so the solution is valid. Thus, the solution is:

\[ \boxed{1} \]

give me the right answer

Let's revisit the equation \( 16^{x-1} = 1024^{x-1} \) and solve it step by step correctly.

We can express both numbers as powers of 2:

Substituting these into the original equation, we have:

\[ (2^4)^{x-1} = (2^{10})^{x-1} \]

This simplifies to:

\[ 2^{4(x-1)} = 2^{10(x-1)} \]

Since the bases are the same, we can set the exponents equal to each other:

\[ 4(x-1) = 10(x-1) \]

To proceed, we can factor out \( (x-1) \):

\[ (x-1)(4 - 10) = 0 \]

This gives us:

\[ (x-1)(-6) = 0 \]

Setting each factor to zero, we find:

\( x - 1 = 0 \) ⟹ \( x = 1 \)
The other equation does not provide any additional solutions since it simplifies to 0.

So the only solution we have from the first factor is:

\[ x = 1 \]

Thus, the final solution is:

\[ \boxed{1} \]

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