<Question>
For what value of $x$ is $2^3\cdot3^x=72$?
</Question>

<think>
<Plan_1>
Okay, so I need to find the value of x that makes the equation 2 cubed times 3 to the power of x equal to 72. Let me start by writing down the equation again to make sure I have it right: 2³ · 3ˣ = 72. Alright, let's break this down step by step.

First, I know that 2³ is 8 because 2 multiplied by itself three times is 2 × 2 × 2, which equals 8. So substituting that into the equation, it becomes 8 · 3ˣ = 72.

Hmm, so if I divide both sides of the equation by 8, that should isolate the term with x in it.
Let me try that: dividing both sides by 8 gives me 3ˣ = 72 / 8.

Calculating the right side, 72 divided by 8 is 9. So now the equation is 3ˣ = 9.

Now, I need to find x such that 3 raised to that power equals 9. I remember that 3 squared is 9 because 3 × 3 = 9. So x must be 2.

Wait, let me check that again. If x is 2, then 2³ is 8, 3² is 9, and multiplying them together gives 8 × 9 = 72, which matches the original equation.
So that seems correct.

But maybe I should approach this problem using logarithms to confirm. Taking the logarithm of both sides could help solve for x. Let's try that.

<prompt_1.1> Use logarithms to solve the equation 2³ · 3ˣ = 72 and confirm the solution. </prompt_1.1>

Alternatively, using natural logarithm: ln(3ˣ) = ln(9). Then x = ln(9)/ln(3). Since ln(9) is ln(3²) = 2 ln(3), so x = 2 ln(3)/ln(3) = 2.
-----

Alternatively, maybe using prime factorization of 72. Let's see, 72 can be broken down into prime factors.

<prompt_1.2> Use prime factorization of 72 to confirm the solution. </prompt_1.2>
</Plan_1>

<Execution_1>
<execution_1.1>
Starting with 8 · 3ˣ = 72. Wait, actually, after simplifying to 3ˣ = 9, I could take the logarithm of both sides. Let's say natural log. So ln(3ˣ) = ln(9). Using the logarithm power rule, that becomes x · ln(3) = ln(9). Then solving for x gives x = ln(9)/ln(3). Since 9 is 3 squared, ln(9) is 2 ln(3), so substituting that in, x = 2 ln(3)/ln(3) = 2.
</execution_1.1>

<execution_1.2>
72 divided by 2 is 36, divided by 2 again is 18, again by 2 is 9, which is 3 squared. So 72 = 2³ · 3². Therefore, comparing that to the left side of the equation, which is 2³ · 3ˣ, we can see that x must be 2.
</execution_1.2>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Another way: maybe express 72 as a product of powers of 2 and 3. Let's factorize 72.

<prompt_2.1> Express 72 as a product of powers of 2 and 3 and compare to the left side of the equation. </prompt_2.1>
</Plan_2>

<Execution_2>
<execution_2.1>
72 divided by 2 is 36, divided by 2 is 18, divided by 2 is 9, which is 3². So 72 = 2³ · 3². Therefore, the equation 2³ · 3ˣ = 72 becomes 2³ · 3ˣ = 2³ · 3². Since the bases are the same and the equation holds, the exponents must be equal. Therefore, x must be 2.
</execution_2.1>
</Execution_2>

<Final_answer>
Thus, the value of $x$ for which $2^3\cdot3^x=72$ is $\boxed{2}$.
</Final_answer>
</think>