Comparing Exponential Values: When Is 89 Greater Than 98?

Many times, we encounter situations where we need to compare exponential values. In this article, we will explore how to determine if 89 is greater than 98. We will present both direct calculation and logarithmic methods to solve this problem accurately and efficiently.

Direct Calculation

To directly calculate the values, we start by expressing 89 and 98 in terms of simpler exponential forms:

For 89:

89 23^9 227 134,217,728

For 98:

98 32^8 316 430,467,216

By comparing the values, it is clear that:

89 134,217,728 8. Therefore, 89 is not greater than 98.

Using Logarithms for Comparison

To compare 89 and 98 using logarithms, we take the logarithm of both expressions:

log 89 9 log 8

log 98 8 log 9

We need to compare 9 log 8 and 8 log 9. To do this, we rearrange the expression as follows:

9 log 8 / 8 log 9 or log 8 / log 9

We need approximate values for log 8 and log 9:

log 8 3 log 2 ≈ 3 × 0.301 0.903

log 9 2 log 3 ≈ 2 × 0.477 0.954

Calculating:

9 log 8 / 8 log 9 ≈ (9 × 0.903) / (8 × 0.954) ≈ 8.127 / 7.632 ≈ 1.064

This indicates that 8 log 9 > 9 log 8, which means:

89 > 98

Another Example: 898 vs. 989

Let's now explore another example: which is larger between 898 and 989. First, we simplify the expressions:

For 898:

898 238 227

For 989:

989 329 316

Comparing these values by comparing the logarithms of these numbers, since the function log x is increasing:

log 898 8 log 89 8 log 23 27 log 2

log 989 9 log 98 9 log 32 16 log 3

We need to compare 27 log 2 and 16 log 3. Calculating approximate values:

log 2 ≈ 0.3010

log 3 ≈ 0.4771

Calculating:

27 log 2 ≈ 27 × 0.3010 8.127

16 log 3 ≈ 16 × 0.4771 7.617

Since 27 log 2 > 16 log 3, we have:

898 > 989