Understanding Exponent Comparison: A Comprehensive Guide
Comparing numbers with large exponents and different bases can seem daunting at first, especially when dealing with values like 3^{210} and 17^{140}. However, utilizing logarithms can simplify the process and allow us to handle such comparisons more effectively. This article delves into the methods and steps required to compare such numbers, ensuring a clear understanding and practical application of the concept.
Introduction to Exponent Comparison
Exponent comparison involves comparing the values of two mathematical expressions with different bases raised to different powers. The challenge lies in managing the size of the numbers involved without performing extensive calculations. One effective method is to use logarithms, which can transform the problem into a more manageable form.
Steps to Compare Numbers with Large Exponents and Different Bases
The process of comparing 3^{210} and 17^{140} using logarithms involves several steps, each critical for a thorough understanding of the comparison.
Step 1: Taking the Logarithm of Both Sides
The first step involves taking the logarithm of both expressions. Using natural logarithms, which are commonly used, we can simplify the problem. We start by expressing the logarithms as follows:
ln(3^{210}) and ln(17^{140})
Step 2: Applying the Power Rule of Logarithms
The power rule of logarithms, often represented as ln(a^b) b ? ln(a), allows us to simplify the expressions. Applying this rule, we get:
ln(3^{210}) 210 ? ln(3)
ln(17^{140}) 140 ? ln(17)
Step 3: Calculating the Logarithms
Using approximate values for the natural logarithms, we can calculate the above expressions. Here are the approximate values:
ln(3) ≈ 1.0986 ln(17) ≈ 2.8332Substituting these values into our expressions, we get:
210 ? ln(3) ≈ 210 ? 1.0986 ≈ 230.826
140 ? ln(17) ≈ 140 ? 2.8332 ≈ 396.648
Step 4: Comparison
By comparing the values, we can determine which expression is larger:
230.826
Since 210 ? ln(3) , it follows that:
3^{210}
Alternative Methods and Examples
There are other methods to compare such numbers, especially when using a calculator is not allowed. Here are a couple of examples.
Example 1: Converting Bases to 10
One method is to convert the bases to base 10 and then compare the exponents:
3^{210} 10^{210 log(3)} 17^{140} 10^{140 log(17)}Using approximate logarithms:
log(3) ≈ 0.4771 log(17) ≈ 1.2304Therefore:
210 log(3) ≈ 210 ? 0.4771 ≈ 100.195 140 log(17) ≈ 140 ? 1.2304 ≈ 172.256Clearly, since 100.195 , it follows that:
3^{210}
Example 2: Equating Bases or Exponents
Another approach is to find a way to equate either the base or the powers of the two numbers. For instance:
3^{210} 27^{70} 17^{140} 289^{70}Since 289 > 27, it follows that:
289^{70} > 27^{70}
Thus:
17^{140} > 3^{210}
Conclusion
The process of comparing numbers with large exponents and different bases using logarithms is a powerful tool. This method simplifies the comparison by reducing it to comparing the products of the logarithms with their respective powers. As demonstrated, both using logarithms and other methods like base conversion or equating bases can lead to the same conclusion.