A comprehensive guide on the laws of logarithms for CBSE Class 11 Applied Mathematics, covering product, quotient, power, and base-change properties with step-by-step solutions.
Logarithmic functions simplify complex multiplications and divisions into additions and subtractions. The Product Law states that the log of a product is the sum of the logs of its factors. The Quotient Law states that the log of a quotient is the difference between the log of the numerator and the denominator.
• Apply the product rule: log_2(8) + log_2(16)
• Identify that 8 = 2^3 and 16 = 2^4
• Evaluate the logs: 3 + 4
• Calculate final sum: 7
Answer: 7
The Power Law allows exponents inside a logarithm to be moved as a multiplier in front of the log. The Change of Base Law is critical when calculating logarithms with different bases, allowing conversion to a common base like 10 or 'e'.
• Apply power law: 2 * log_3(81)
• Express 81 as 3^4: 2 * log_3(3^4)
• Bring exponent 4 to the front: 2 * 4 * log_3(3)
• Since log_3(3) = 1, calculate 8 * 1
Answer: 8