Logarithmic and Exponential Functions Video Lecture | Mathematics (Maths) Class 12 - JEE

Ans. A logarithmic function is the inverse of an exponential function. It represents the relationship between a logarithm and its base. In simple terms, it calculates the exponent to which a base must be raised to obtain a certain value. The general form of a logarithmic function is y = logₐ(x), where x is the input value, a is the base, and y is the corresponding output value.

Ans. Logarithmic functions and exponential functions are closely related, but they have distinct differences. While an exponential function involves raising a base to an exponent, a logarithmic function involves finding the exponent to which a base must be raised to obtain a certain value. In other words, exponential functions represent exponential growth or decay, whereas logarithmic functions represent the inverse relationship of exponentiation.

Ans. Logarithmic functions find various applications in the real world. Some examples include: 1. pH scale: The pH scale, used to measure the acidity or alkalinity of a substance, is based on logarithmic functions. Each unit change on the pH scale represents a tenfold difference in acidity or alkalinity. 2. Sound intensity: The decibel scale, which measures the intensity of sound, uses logarithmic functions. Each increase of 10 decibels corresponds to a tenfold increase in sound intensity. 3. Richter scale: The Richter scale, used to measure the magnitude of earthquakes, is based on logarithmic functions. Each increase of one unit on the Richter scale corresponds to a tenfold increase in the earthquake's strength.

Ans. An exponential function is a mathematical function in which a constant base is raised to a variable exponent. It represents exponential growth or decay, where the rate of change is proportional to the current value. The general form of an exponential function is y = a * b^x, where x is the input value, b is the base, a is a constant multiplier, and y is the corresponding output value.

Ans. Exponential functions play a crucial role in finance and compound interest calculations. They are used to model the growth of investments or loans over time. For example, the compound interest formula A = P(1 + r/n)^(nt) utilizes exponential functions, where A represents the final amount, P is the principal investment, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the number of years. This formula allows individuals and businesses to estimate the future value of their investments or the amount owed on a loan.