In common (base 10) logarithms, what is the characteristic of log 0.0000134?

Introduction / Context:
This question focuses on the concept of the characteristic of a common (base 10) logarithm. For numbers less than 1, the characteristic is negative and tells us how far the number is from 1 in terms of powers of 10. Understanding this is essential for reading and using logarithm tables correctly.

Given Data / Assumptions:

• We are working with common logarithms (base 10).
• The number is 0.0000134.
• We are asked only for the characteristic, not the mantissa.

Concept / Approach:
For a positive number N, the common logarithm log₁₀ N can be written as C + M, where C is the characteristic (integer part) and M is the mantissa (fractional part). For N between 0 and 1, the characteristic is negative. A useful rule is: if the first non-zero digit after the decimal occurs in the (k + 1)th place, then the characteristic is -(k + 1). Alternatively, rewrite the number in scientific notation N = a × 10^n, where 1 ≤ a < 10, and then log N = n + log a; here n will be the characteristic if a is between 1 and 10.

Step-by-Step Solution:
Write 0.0000134 in scientific notation. We can move the decimal point 5 places to the right: 0.0000134 = 1.34 × 10^-5. Then log(0.0000134) = log(1.34 × 10^-5). Using log(ab) = log a + log b, this becomes log 1.34 + log 10^-5. log 10^-5 = -5, and log 1.34 is a positive fraction between 0 and 1. Therefore log(0.0000134) = -5 + (a fractional mantissa). The characteristic is the integer part, which is -5.

Verification / Alternative check:
We can also count the zeros after the decimal point before the first non-zero digit. In 0.0000134, after the decimal point there are four zeros before the 1. The first significant digit 1 is in the fifth place after the decimal. By the rule, characteristic = -(number of digits to the first significant digit) = -5. This matches the previous method.

Why Other Options Are Wrong:
5: This would correspond to a number around 10^5, which is huge compared to 0.0000134, which is very small.
6 and -6: These would indicate the number is closer to 10^6 or 10^-6. But 0.0000134 is equal to 1.34 × 10^-5, so the power of 10 involved is -5, not -6 or 6.

Common Pitfalls:
Students often miscount the zeros or confuse the sign for numbers less than 1. Another mistake is to think that the characteristic is simply the number of zeros, not the number of digits to the first non-zero digit. Always check by converting to scientific notation and identifying the power of 10 explicitly.

Final Answer:
The characteristic of log 0.0000134 is -5.