<Question>
If $10^x - 10 = 9990,$ what is $x$ equal to?
</Question>

<think>
<Plan_1>
Okay, let's see. I need to solve for x in the equation 10^x minus 10 equals 9990. So, the equation is 10^x - 10 = 9990. Hmm, right. Let me start by recalling how to handle exponential equations.

First, maybe I should isolate the term with the exponent. That usually helps. So, if I add 10 to both sides of the equation, that should get rid of the -10 on the left side.
Let's try that: 10^x - 10 + 10 = 9990 + 10. Simplifying both sides, that becomes 10^x = 10000.

Alright, so now the equation is 10^x = 10000. I need to find x such that 10 raised to the power of x equals 10000.

Wait, 10000 is a power of 10, right? Let me think. 10^1 is 10, 10^2 is 100, 10^3 is 1000, and 10^4 is 10000.
So, 10^4 = 10000. Therefore, x must be 4.

Is that the answer? Let me check.

Plugging x = 4 back into the original equation: 10^4 - 10 = 10000 - 10 = 9990.

But wait, maybe I should consider if there are any other solutions. Since the base 10 exponential function is one-to-one, meaning that for each y > 0, there's only one x such that 10^x = y. So, since 10000 is positive, there's only one real solution here, which is x = 4.

Alternatively, if I use logarithms to solve for x, maybe that's another way.

Taking the logarithm base 10 of both sides: log_10(10^x) = log_10(10000). The left side simplifies to x because log_b(b^x) = x.

<prompt_1.1> Solve the equation using logarithms as an alternative method. </prompt_1.1>

But just to make sure I didn't skip any steps or make any mistakes in my reasoning. Let's go through it again.

<prompt_1.2> Review the solution to ensure no steps were skipped or mistakes made. </prompt_1.2>

Alternatively, if I didn't recognize that 10000 is 10^4, maybe I could factor 10000 into primes.

<prompt_1.3> Solve the equation by factoring into primes. </prompt_1.3>
</Plan_1>

<Execution_1>
<execution_1.1>
And the right side is log_10(10000). Since 10000 is 10^4, log_10(10^4) is 4. Therefore, x = 4.
</execution_1.1>

<execution_1.2>
Original equation: 10^x - 10 = 9990. Adding 10 to both sides gives 10^x = 10000. Then, since 10000 is 10^4, x is 4.
</execution_1.2>

<execution_1.3>
Let's see, 10000 is 10^4, which is (2*5)^4 = 2^4 * 5^4. But 10^x is 2^x * 5^x. So, setting 2^x * 5^x = 2^4 * 5^4. Since the prime factors must be equal, x must be 4. That also confirms it. So, even if I approach it that way, x is 4.
</execution_1.3>
</Execution_1>

<Final_answer>
Therefore, the value of $x$ is $\boxed{4}$.
</Final_answer>
</think>