A comprehensive guide on Common and Natural logarithms, covering definitions, key properties, and step-by-step solved examples for CBSE Class 11 Applied Mathematics.
If a and N are positive real numbers and a is not equal to 1, then the logarithm of a number N to the base a is the exponent x to which a must be raised to obtain N. It is denoted as log_a(N) = x, which is equivalent to the exponential form a^x = N.
• Express the equation in logarithmic form: log_2(16) = x
• Convert to exponential form: 2^x = 16
• Express 16 as a power of 2: 2^x = 2^4
• Equate the exponents: x = 4
Answer: log_2(16) = 4
A common logarithm is a logarithm with base 10. It is often written without explicitly stating the base, such as log(x) or log_10(x). Common logarithms are primarily used in scientific calculations and numerical analysis involving the decimal number system.
• Assume base 10: log_10(1000) = x
• Convert to exponential form: 10^x = 1000
• Express 1000 as a power of 10: 10^x = 10^3
• Equate the exponents: x = 3
Answer: log(1000) = 3
A natural logarithm is a logarithm with base 'e', where 'e' is Euler's constant (approximately 2.71828). It is denoted as ln(x) or log_e(x). Natural logarithms are vital in calculus and describing growth or decay processes in nature.
• Set the expression to x: ln(e^5) = x
• Convert to exponential form with base e: e^x = e^5
• Since the bases are identical, compare the exponents: x = 5
Answer: ln(e^5) = 5
Logarithms follow specific algebraic rules that simplify complex products, quotients, and powers into simpler operations.
• Apply the product rule: log(M) + log(N) = log(MN)
• Combine the arguments: log(50 * 2)
• Calculate the product: log(100)
• Solve for base 10: log_10(10^2) = 2
Answer: The result is 2