If the logarithm of a number is $-3.153$, what are characteristic and mantisa?

### Concept & Formula Every logarithm consists of two distinct parts: an integer part called the **characteristic**, and a fractional/decimal part called the **mantissa**. The fundamental rule of standard logarithms is that the mantissa must always be a non-negative value ($0 \le \text{mantissa} < 1$). If a logarithm is a negative number, you must restructure it so the decimal part becomes positive. ### Step-by-Step Solution Given the logarithm value: $-3.153$ Step 1: Separate the number into its integer and decimal components. $$ -3.153 = -3 - 0.153 $$ Step 2: Notice that the decimal part ($-0.153$) is negative, which violates the rule for mantissas. To make it positive, we subtract $1$ from the integer part and add $1$ to the decimal part. This doesn't change the overall mathematical value of the expression. $$ -3.153 = (-3 - 1) + (1 - 0.153) $$ Step 3: Simplify both parts. The integer part (characteristic) becomes: $-3 - 1 = -4$ The decimal part (mantissa) becomes: $1 - 0.153 = 0.847$ So, the characteristic is $-4$ (often written as $\bar{4}$), and the mantissa is $0.847$. ### Exam Strategy & Shortcut **The "Next Lowest Integer" Rule:** When dealing with a negative logarithm, the characteristic is always the next integer that is *lower* (more negative) than the given number. For $-3.153$, the next lowest integer is $-4$. That is your characteristic instantly. To find the mantissa, just find the complement of the decimal part to $1$. ($1 - 0.153 = 0.847$). ### Common Pitfall The most common trap is simply taking the number exactly as it is written, leading students to choose Characteristic $= -3$ and Mantissa $= -0.153$. Remember that standard log tables and calculations require a positive mantissa. ### Final Answer **Therefore, the correct answer is Characteristic $= -4$, mantissa $= 0.847$.**