<Question>
What power of 4 is equal to 8? Express your answer as a common fraction.
</Question>

<think>
<Plan_1>
Okay, let's see. The question is asking, "What power of 4 is equal to 8? Express your answer as a common fraction." Hmm, so I need to find an exponent that when applied to 4 gives 8. Let me start by writing this out in mathematical terms.
So, if I let x be the exponent, then the equation would be 4^x = 8.

Alright, now I need to solve for x here. Both 4 and 8 are powers of 2, right?
4 is 2 squared, and 8 is 2 cubed. Maybe I can rewrite the equation with base 2 instead of base 4. Let me try that.

<prompt_1.1> Rewrite the equation 4^x = 8 using base 2 as described in the above plan. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
So, 4 is 2^2, so 4^x would be (2^2)^x. And that's equal to 8, which is 2^3. So substituting those in, the equation becomes (2^2)^x = 2^3.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Now, when you raise a power to another power, you multiply the exponents.
So, (2^2)^x is 2^(2x). So now the equation is 2^(2x) = 2^3.

Since the bases are the same (both base 2), the exponents must be equal. Therefore, 2x = 3.
To solve for x, divide both sides by 2, so x = 3/2.

Wait, but the question says to express the answer as a common fraction.

<prompt_2.1> Verify the solution x = 3/2 by substituting it back into the original equation 4^x = 8. </prompt_2.1>

Wait a second, but let me make sure I didn't make a mistake here. Let me think again.
Sometimes with exponents, especially fractional ones, things can get tricky. Let me verify each step.

<prompt_2.2> Re-evaluate each step of the solution to ensure there are no mistakes. </prompt_2.2>

Is there another way to approach this problem? Maybe using logarithms? Let's try that method to confirm.

<prompt_2.3> Solve the equation 4^x = 8 using logarithms as described in the above plan. </prompt_2.3>

Alternatively, using natural logarithm: ln(4^x) = ln(8). Which is x*ln(4) = ln(8). Then x = ln(8)/ln(4).
Calculating that, ln(8) is ln(2^3) = 3*ln(2), and ln(4) is ln(2^2) = 2*ln(2). So x = (3*ln(2))/(2*ln(2)) = 3/2. Same result.

Wait, but maybe there's a way to think about this without changing the base. Let's see.

<prompt_2.4> Consider solving the equation 4^x = 8 without changing the base as described in the above plan. </prompt_2.4>
</Plan_2>

<Execution_2>
<execution_2.1>
3/2 is already a common fraction, so that should be the answer. But let me double-check to make sure I didn't make a mistake. Let's substitute x = 3/2 back into the original equation and see if it holds true.
</execution_2.1>

<execution_2.2>
Starting with 4^x = 8. Expressing 4 as 2^2 and 8 as 2^3: correct. Then (2^2)^x = 2^(2x), and 2^3 is correct. Then setting the exponents equal: 2x = 3. Solving for x gives 3/2.
</execution_2.2>

<execution_2.3>
If I take the logarithm of both sides of the equation 4^x = 8. Let's use natural log for example. So ln(4^x) = ln(8). Then using the logarithm power rule, that becomes x*ln(4) = ln(8). Then x = ln(8)/ln(4). Let me compute that. Since 8 is 2^3 and 4 is 2^2, ln(8) is 3*ln(2) and ln(4) is 2*ln(2). So again, x = (3*ln(2))/(2*ln(2)) = 3/2.
</execution_2.3>

<execution_2.4>
If 4^x = 8, then I can take the square root of both sides or something? Let me see. If I take the square root, that would be 4^(x/2) = 8^(1/2). Hmm, not sure if that helps. Alternatively, maybe express 8 as 4^(something). Since 4^1 = 4, 4^2 = 16, which is too big. So 8 is between 4^1 and 4^2, so the exponent must be a fraction.
</execution_2.4>
</Execution_2>

<Plan_3>
Based on execution_2.1:
So 4^(3/2). Let's compute that. The exponent 3/2 can be thought of as taking the square root of 4 first, which is 2, and then raising it to the 3rd power.
So, (sqrt(4))^3 = 2^3 = 8. Yep, that works. Alternatively, 4^(3/2) = (4^(1/2))^3 = (2)^3 = 8. So that checks out.

Wait, but another way to check is to compute 4^(3/2). Let's do that step by step.

<prompt_3.1> Verify the solution by calculating 4^(3/2) step by step. </prompt_3.1>
</Plan_3>

<Execution_3>
<execution_3.1>
The exponent 3/2 can be split into 4^(3 * (1/2)). So first, take the square root of 4, which is 2, and then cube it. 2^3 is 8. Yep, that's correct.

Alternatively, if I use fractional exponents, 4^(3/2) is the same as exp((3/2)*ln(4)). Let's compute that numerically. ln(4) is approximately 1.386. So (3/2)*1.386 ≈ 2.079. Then exp(2.079) is approximately 8.
</execution_3.1>
</Execution_3>

<Plan_4>
Based on execution_3.1:
I think that's thorough. Let me just recap to make sure I didn't skip any steps.
The problem is to find x such that 4^x = 8. Both 4 and 8 are powers of 2, so rewrite them as 2^2 and 2^3 respectively.

<prompt_4.1> Recap and verify the solution using different methods as described in the above plan. </prompt_4.1>
</Plan_4>

<Execution_4>
<execution_4.1>
Then set the exponents equal after equating the bases. Solve for x, which gives 3/2. Checked using logarithms and by direct substitution. Also verified by calculating 4^(3/2) step by step. All methods confirm that x = 3/2 is correct. Therefore, the answer is 3/2.
</execution_4.1>
</Execution_4>

<Final_answer>
Therefore, the power of 4 that equals 8 is $\boxed{\frac{3}{2}}$.
</Final_answer>
</think>