Comparing Exponential Expressions: Which is Larger, 57 or 75?

When faced with comparing exponential expressions like 57 and 75, several methods can help us determine which is larger quickly and efficiently. These methods include direct calculation, logarithms, and properties of exponents. Let's explore these techniques in detail.

Direct Calculation

One straightforward approach is to calculate both values directly:

57: 5 × 5 × 5 × 5 × 5 × 5 × 5 78,125 75: 7 × 7 × 7 × 7 × 7 16,807

By comparing these results, we can see that 57 (78,125) is larger than 75 (16,807).

Using Logarithms

For quick mental calculations, taking the logarithm of both expressions can be helpful. Let's use logarithms to compare the two expressions:

Calculate (7 log 5) and (5 log 7): Since (log 5 0.7030) and (log 7 0.8451), we have: 7 × 0.7030 4.9210 5 × 0.8451 4.2255

Since (4.9210 > 4.2255), we can conclude that 57 > 75.

Comparing Bases

Another neat trick involves comparing the bases with the number (e) (approximately 2.718). The number with the base closer to (e) is generally the larger exponentiation expression:

Compare the bases: The base for 57 is 5. The base for 75 is 7.

Since 5 is closer to (e) (2.718) than 7, 57 is larger than 75.

Logarithmic Properties

We can also use logarithmic properties to solve the problem:

Apply logarithms: (log(5^7) 7 log 5) (log(7^5) 5 log 7)

Using approximate values:

(7 log 5 7 times 0.7030 4.9210) (5 log 7 5 times 0.8451 4.2255)

Since (log(5^7) > log(7^5)), 57 is larger than 75.

Breaking Down the Expressions

An alternative method involves breaking down the expressions into parts:

Break down the exponents: 57 can be written as (5^4 times 5^3) 75 can be written as (7^3 times 7^2)

Now compare the parts:

(5^4 625) (5^3 125) (7^3 343) (7^2 49)

Each part of 57 is larger than the corresponding part of 75:

625 > 343 125 > 49

Therefore, 57 is greater than 75.

Conclusion

In summary, 57 (78,125) is larger than 75 (16,807). This can be determined through direct calculation, logarithms, properties of bases, or breaking down the expressions. Each method has its own utility depending on the context and time constraints.

Understanding these techniques can help in solving similar problems quickly and efficiently, making it a valuable skill for competitive aptitude tests and various practical applications.