Okay, let's see. I need to find the value of x for which 2 cubed times 3 to the power of x equals 72. Hmm. Let me start by writing down the equation to make it clear: 2³ · 3ˣ = 72. 

First, I remember that 2³ is 8. So substituting that in, the equation becomes 8 · 3ˣ = 72. Now, I need to solve for x. To isolate 3ˣ, I should divide both sides of the equation by 8. Let me check that. If I divide both sides by 8, then 8 divided by 8 is 1, so 3ˣ = 72 / 8. Calculating 72 divided by 8... 8 times 9 is 72, right? So 72 divided by 8 is 9. Therefore, 3ˣ = 9.

Now, I need to find x such that 3 to the power of x equals 9. Hmm, 3 squared is 9, so x must be 2. Let me verify that. Plugging x = 2 back into the original equation: 2³ · 3² = 8 · 9 = 72. Yep, that works. 

Wait, but let me make sure I didn't skip any steps. Maybe I should check if there's another way to approach this problem. For example, using logarithms. If I take the logarithm of both sides, maybe that could help. Let's try that. 

Starting again with 2³ · 3ˣ = 72. Taking the natural logarithm of both sides: ln(2³ · 3ˣ) = ln(72). Using logarithm properties, this becomes ln(2³) + ln(3ˣ) = ln(72). Which simplifies to 3 ln 2 + x ln 3 = ln 72. Then, solving for x: x ln 3 = ln 72 - 3 ln 2. Therefore, x = (ln 72 - 3 ln 2) / ln 3. 

Let me compute that. First, ln 72. Let me recall that ln 72 is the natural log of 72. Similarly, ln 2 and ln 3 are known constants. But maybe I can simplify this expression. Let's see. 72 is 8 times 9, which is 2³ · 3². So ln 72 = ln(2³ · 3²) = 3 ln 2 + 2 ln 3. Therefore, substituting back into the expression for x: x = (3 ln 2 + 2 ln 3 - 3 ln 2) / ln 3. The 3 ln 2 terms cancel out, leaving x = 2 ln 3 / ln 3 = 2. So that's the same answer as before. 

Okay, so both methods give x = 2. That seems consistent. Let me check if there's another way to factor 72. 72 divided by 8 is 9, which is 3 squared. So another way to see it: 2³ · 3ˣ = 2³ · 3². Therefore, comparing exponents, x must be 2. 

Is there any possibility that there's another solution? For example, if x were a different number, would the equation still hold? Let's test x = 1: 2³ · 3¹ = 8 · 3 = 24, which is less than 72. x = 3: 8 · 27 = 216, which is way more than 72. So x has to be between 1 and 3. Since 3² is 9, which works, and 3³ is 27, which is too big. So x must be 2. 

Alternatively, if I didn't recognize that 72 is 8 times 9, maybe I could factor 72 into its prime factors. Let's do that. 72 divided by 2 is 36, divided by 2 again is 18, divided by 2 again is 9. So 2³ · 3². Therefore, 2³ · 3ˣ = 2³ · 3² implies x = 2. 

So, all approaches lead to x = 2. I think that's solid. Let me just recap to make sure I didn't miss anything. The original equation is 2³ times 3 to the x equals 72. Breaking down 72 into prime factors gives 2³ times 3², which directly shows that x must be 2. Alternatively, solving algebraically by dividing both sides by 8 gives 3ˣ = 9, which is 3 squared, so x is 2. Using logarithms also confirms that x is 2. 

I don't see any alternative solutions here. Since 3ˣ is an exponential function, it's one-to-one, so there's only one solution. Therefore, the answer is definitely 2. 

Just to be thorough, let's plug x = 2 back into the original equation again. 2³ is 8, 3² is 9. Multiplying them gives 8 * 9 = 72, which matches the right-hand side. Yep, that checks out. 

I think that's all there is to it. No steps seem missing, and all methods converge to the same answer. So I'm confident that x = 2 is the correct value.
Therefore, the value of $x$ that satisfies the equation is $\boxed{2}$.
</think>