What is the characteristic (integer part) of the common logarithm of the number 0.0000134?

Introduction / Context:
This question tests the concept of the characteristic of a common logarithm. The characteristic is the integer part of log10 N, and it indicates the order of magnitude of the number in base 10. Understanding how to find the characteristic quickly, especially for numbers less than 1 and greater than 0, is important for working with logarithm tables and for solving digit and magnitude problems in aptitude exams.

Given Data / Assumptions:
- The number is 0.0000134.
- We are dealing with the common logarithm, that is log10 of the number.
- The characteristic is defined as the integer part of the logarithm, which will be negative for numbers between 0 and 1.

Concept / Approach:
For any positive number N, you can write it in scientific notation as N = m * 10^k, where 1 less than or equal to m less than 10 and k is an integer. Then log10 N = log10 m + k. The characteristic is the integer k when log10 m is between 0 and 1. For numbers greater than 1, k is non negative, and for numbers between 0 and 1, k is negative. The rule for a number between 0 and 1 is that if N lies between 10^(-n) and 10^(-n+1), then the characteristic is -n.

Step-by-Step Solution:
Write the number 0.0000134 in the form m * 10^k. Count the number of decimal places until the first non zero digit. The number 0.0000134 has 5 zeros after the decimal point before the first non zero digit 1. We can rewrite 0.0000134 = 1.34 * 10^-5. Now compute log10 (0.0000134) as log10 (1.34 * 10^-5). Use the product rule: log10 (1.34 * 10^-5) = log10 1.34 + log10 10^-5. We know log10 10^-5 = -5. The value log10 1.34 is between 0 and 1, so it does not change the integer part. Thus log10 (0.0000134) = (a small positive number) + (-5) = something between -5 and -4. Therefore the characteristic, which is the integer part of the logarithm, is -5.

Verification / Alternative check:
We can use the general rule for numbers between 0 and 1. A number of the form 0.0000...x, where there are n zeros after the decimal point before the first non zero digit, has a characteristic of -(n + 1). In this case, there are 4 zeros after the decimal point before the first non zero digit if we count carefully: 0.0000134 can be seen as 0.0000 134 with four zeros after the decimal point before 1, which gives characteristic -5. This shortcut is commonly used when working quickly with log tables.

Why Other Options Are Wrong:
5: This would be the characteristic of a large number near 10^5, not a small number less than 1.
6: This would correspond to an even larger number of the order of 10^6, which is completely inconsistent with 0.0000134.
-6: This would indicate a much smaller number of order 10^-6 or less, whereas 0.0000134 is between 10^-5 and 10^-4, not as small as 10^-6.

Common Pitfalls:
A frequent mistake is to miscount the number of zeros after the decimal point or to think that the number of zeros directly gives the characteristic without adding or subtracting one. Another error is to confuse the characteristic with the mantissa. The mantissa is the fractional part of the logarithm and is always non negative when using standard logarithm tables. Here we only need the integer part, which is negative for numbers between 0 and 1. Carefully writing the number in scientific notation helps avoid these errors.

Final Answer:
The characteristic of log10 (0.0000134) is -5.