Final answer:
To determine what '4' equates to in the context of powers of 10, we need to understand integer powers and exponentiation. The power 10⁴ equals 10,000, and 10² equals 100. However, there is no integer power of 10 that results in 4, which means we'd use logarithmic functions for an exact result.
Step-by-step explanation:
If 10⁴ equals 10,000, it implies that the power of 4 to the base 10 is equal to 10,000. In mathematics, we understand this concept through integer powers, where something like 10³ is 10 multiplied by itself three times, resulting in 1,000. To find out what '4' is equal to in this context, we need to realize that 10² designates 10 squared, or 10 × 10, which equals 100. Therefore, identifying the power of 10 that gives us '4' is simply a matter of recognizing the pattern or applying the basics of exponentiation.
The sure-fire way to understand integer powers is by example: 10³ (10 cubed) is 10 × 10 × 10, or 1,000. Similarly, 10⁻⁴ (10 to the negative fourth power) is four repeated (multiplied) instances of 10⁻¹, each one being 0.1. When these 0.1's are multiplied together, we end up with 0.0001 or 10⁻⁴, which is 1/10,000. Conversely, to find the power of 10 that results in 4, we need to find an exponent such that 10 raised to that exponent equals 4. As there is no integer exponent that will result in exactly 4, we can only approach it through a logarithmic function or decimal exponentiation, which is beyond basic integer powers.