Comparing the Values of 103 and 310

When faced with the task of comparing 10^3 and 3^{10}, understanding the respective values and applying basic mathematical techniques can provide clarity. This article delves into the methods and results of this comparison, offering insights into exponentiation and logarithmic analysis.

Direct Calculation

One straightforward approach to comparing these values is through direct calculation.

Calculating 103:

10^3 1000

Calculating 310:

This involves multiplying 3 by itself 10 times. We can break this down using exponents and multiplication:

- 3^2 9

- 3^3 27

- 3^4 81

- 3^5 243

- 3^6 729

- 3^7 2187

3^{10} 3^7 * 3^3 2187 * 27

Calculating 3^{10} gives us: 3^{10} 59049

Logarithmic Analysis

For a deeper understanding, we can use logarithms. Taking the logarithm (base 10) of each expression can simplify the comparison:

For 103:

Logarithm: log(10^3) 3

For 310:

Logarithm: log(3^{10}) 10 * log(3) ≈ 4.771

Here, we see that 10^{3} has a lower logarithmic value compared to 3^{10}, indicating that 3^{10} is much larger.

Intuitive Comparison

From a more intuitive perspective, when both bases and exponents are greater than the base of the natural logarithm (e ≈ 2.718), the exponent has a greater impact on the value. Thus, for larger exponents, the expression with the smaller base but the larger exponent will generally dominate. Given that 10 and 3 are both greater than e, the exponent's effect makes 3^{10} larger than 10^3. Additionally, 3^3 27, which is still greater than 10, leading to the conclusion that 3^9 will be much larger than 10^3. Therefore, 3^{10} will also be much larger than 10^3.

Conclusion

In conclusion, through direct calculation, logarithmic analysis, and intuitive reasoning, it is clear that 3^{10} is significantly larger than 10^3. This example demonstrates the power of exponentiation and provides a useful tool for understanding large numbers and their relationships.