1. Without using a calculator, evaluate the following: (i) \(\log_2 18 – \log_2 9\) Using the logarithmic property: \(\log_b x – \log_b y = \log_b \frac{x}{y}\): \[\log_2 18 – \log_2 9 = \log_2 \frac{18}{9} = \log_2 2 = 1\] Answer: \(1\) (ii) \(\log_2 64 + \log_2 2\) Using the logarithmic property: \(\log_b x + \log_b…
Logarithms are indispensable tools in mathematics, simplifying complex calculations and making exponential growth and decay problems manageable. The laws of logarithms provide rules to work efficiently with logarithmic expressions, while their applications span diverse fields such as science, engineering, and finance. This article delves into the fundamental laws of logarithms and their real-world uses, helping…
1. Find the characteristic of the following numbers (i) \(5287\) \[\text{Characteristic }= 3\] (ii) \(59.28\) \[\text{Characteristic }= 1\] (iii) \( 0.0567\) \[\text{Characteristic }=\overline{2}\] (iv) \(234.7\) \[\text{Characteristic }= 2\] (v) \(0.000049\) \[\text{Characteristic }=\overline{5}\] (vi) \(145000\) \[\text{Characteristic }= 5\] 2 Find the logarithm of the following numbers (i) \(43\) \[\text{Characteristic }= 1\] \[\text{Mantissa }= 6335\] \[\log(43) =…
Logarithms are powerful tools in mathematics, widely used in science, engineering, and everyday problem-solving. Understanding their key types, such as common and natural logarithms, along with related concepts like mantissa, characteristic, and antilogarithms, is essential for mastering exponential and logarithmic equations. This article provides a detailed yet simple explanation of these terms, complete with formulas…
1. Express each of the following in logarithmic form: (i) \( 10^3 = 1000 \) \[\log_{10} 1000 = 3\] (ii) \( 2^8 = 256 \) \[\log_{2} 256 = 8\] (iii) \( 3^{-3} = \frac{1}{27} \) \[\log_{3} \frac{1}{27} = -3\] (iv) \( 20^2 = 400 \) \[\log_{20} 400 = 2\] (v) \( 16^{-\frac{1}{4}} = \frac{1}{2} \)…
Logarithms are one of the most useful mathematical concepts, simplifying complex calculations and enabling us to solve problems involving exponential growth and decay. From scientific research to everyday applications like pH calculations or measuring sound intensity, logarithms are a fundamental tool. This article provides a clear definition of logarithms, their formulas, and examples to help…
Express the following numbers in scientific notation: (i) \( 2000000 \) \[2000000 = 2.0 \times 10^6\] (ii) \( 48900 \) \[48900 = 4.89 \times 10^4\] (iii) \( 0.0042 \) \[0.0042 = 4.2 \times 10^{-3}\] (iv) \( 0.0000009 \) \[0.0000009 = 9.0 \times 10^{-7}\] (v) \( 73 \times 10^3 \) \[73 \times 10^3 = 7.3 \times…
Logarithms and scientific notation are essential mathematical tools for simplifying complex calculations. Whether you’re dealing with extremely large or small numbers, these concepts allow for easy representation and manipulation. This article explores logarithms, scientific notation, and the processes for converting numbers between ordinary and scientific notation. By mastering these techniques, you can enhance your problem-solving…
The 10th Class Date Sheet 2025 for all Punjab Boards is anticipated to be announced soon. According to the latest official notification, the 9th class exams are scheduled to commence on 5 March 2025. To assist students, Livemath.nt has provided the date sheet, allowing students to view and download the 10th Class Date Sheet 2025…
The 9th Class Date Sheet 2025 for all Punjab Boards is anticipated to be announced soon. According to the latest official notification, the 9th class exams are scheduled to commence on 25 March 2025. To assist students, Livemath.nt has provided the date sheet, allowing students to view and download the 9th Class Date Sheet 2025…
