Question

Characteristic of \(\log 0.5828\) is (a) \(-2\) (b) 1 (c) \(\overline{2}\) (d) \(\bar{I}\)

Short answer

(b) 1

Step-by-step solution

Knowing the Characteristic

The characteristic of a number in logarithm is the integral part of a logarithm. A characteristic of a positive integer is always one less than the number of digits in it. For any number between 0 and 1, the characteristic is given by -m where m is the number of zeros between the decimal and the first non-zero digit.

Identify the Given Number

The given number is 0.5828, which is less than 1. And there are no zeros between the decimal and the first non-zero digit.

Calculating the Characteristic

Using the rule for any number between 0 and 1, the characteristic is -m where m is the number of zeros between the decimal and the first non-zero digit. In case of 0.5828, m=0. So, the characteristic would be -0.

Key concepts

Characteristic of Logarithm

The characteristic of a logarithm helps us understand the integer part of a logarithmic value in its expression. It’s a useful concept, especially when dealing with the logarithms of decimals and integers.For numbers greater than 1, the characteristic is one less than the number of digits to the left of the decimal point. For example, for 523, which has three digits, the characteristic is 2, because it’s one less than the three digits present.Conversely, for numbers between 0 and 1, the characteristic is determined differently. It is \(-m\), where \(m\) is the number of zeros between the decimal point and the first non-zero digit. To illustrate, considering the number 0.0043:

• Count the zeros after the decimal, here it’s 2.
• Therefore, the characteristic would be \(-2\).

Understanding the characteristic is crucial as it aids in simplifying and solving logarithmic expressions by giving insights into the shift and scale of numbers in the log scale.

Logarithmic Properties

Logarithms have a rich set of properties that simplify multiplication, division, and exponentiation problems by turning them into addition, subtraction, and multiplication operations, respectively.

• Product Rule: \(\log_b(xy) = \log_b x + \log_b y\)
• Quotient Rule: \(\log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y\)
• Power Rule: \(\log_b(x^y) = y \cdot \log_b x\)
• Change of Base Formula: Another useful property is changing the base of logs: \(\log_b x = \frac{\log_k x}{\log_k b}\).

These properties facilitate mathematical problem-solving by converting complex multiplicative problems into simpler additive ones. They help in decomposing and combining logarithmic expressions efficiently, making calculation not only easier but often more intuitive.

Mathematical Problem Solving

Solving mathematical problems with logarithms often starts by understanding the problem and identifying relevant properties or rules that apply. When faced with a logarithmic problem:

• Identify and Simplify: Look for opportunities to apply logarithmic properties to simplify the expressions.
• Find Characteristics: Determine the characteristic, which can often give insights into the scale of calculation and simplify steps.
• Use Properties Effectively: Knowing which property to use at the right time, like the product or quotient rules, can make a difference.

By breaking down the problem into smaller manageable parts using logarithm rules, you can navigate through the steps logically. Practicing identifying when to use these properties effectively is crucial to becoming proficient in logarithmic problem-solving.