Question

solve for x

16^x-1=1024^x-1

Answer 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:
- \( 16 = 2^4 \)
- \( 1024 = 2^{10} \)

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:

\[
\boxed{1}
\]

Answer 2

give me the right answer

Answer 3

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:

- \( 16 = 2^4 \)
- \( 1024 = 2^{10} \)

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:

1. \( x - 1 = 0 \) ⟹ \( x = 1 \)

2. 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:

\[
\boxed{1}
\]

If you require anything else or a further breakdown, please let me know!