Comparing the Values of 129, 1011, and 1110: A Comprehensive Analysis

To determine which of the following numbers—129, 1011, or 1110—has the highest value, we can use a logarithm-based approach for an accurate comparison.

Logarithm-Based Analysis

Let's calculate the logarithm of each number. Using the logarithm properties, we have:

1. 129

(log {12^9} 9 cdot log 12)

(log 12 approx 1.0792)

Substituting the value:

(log {12^9} 9 cdot 1.0792 9.7128)

2. 1011

(log {10^{11}} 11 cdot log 10 11)

3. 1110

(log {11^{10}} 10 cdot log 11)

(log 11 approx 1.0414)

Substituting the value:

(log {11^{10}} 10 cdot 1.0414 10.414)

Comparison of Logarithmic Values

From the above calculations, we can compare the logarithmic values:

(log {12^9} approx 9.7128) (log {10^{11}} 11) (log {11^{10}} approx 10.414)

Based on these logarithmic values, we can see that:

1011 has the highest logarithmic value. 1110 has the second-highest logarithmic value. 129 has the lowest logarithmic value.

Therefore, 1011 has the highest value among the three numbers.

Calculating the Actual Values

To further verify the comparison, let's calculate the actual values of the numbers:

129 5,159,780,352 1011 10,000,000,000 (10 billion) 1110 25,937,424,601 (25.93 billion)

From these calculations, it is clear that 1011 indeed has the highest value.

Conclusion

Thus, from the logarithmic analysis and actual value calculations, we can conclude that:

The highest value among 129, 1011, and 1110 is 1011.

Understanding the logarithmic approach can help us efficiently compare large numbers. This method simplifies the comparison and provides a clear understanding of the magnitude of each number. If you have any further questions or need more detailed information, feel free to ask!