Logarithms Video Lecture | General Aptitude for GATE - Mechanical Engineering

Ans. A logarithm is the inverse operation of exponentiation. It represents the power to which a base number must be raised to obtain a given number. In other words, if we have the equation x = b^y, then the logarithm of x to the base b is written as y = log_b(x). Logarithms are commonly used in mathematics, science, and engineering to solve equations involving exponential growth or decay.

Ans. Logarithms have several important properties that make them useful in calculations. Some of the main properties of logarithms include: - The product rule: log_b(xy) = log_b(x) + log_b(y) - The quotient rule: log_b(x/y) = log_b(x) - log_b(y) - The power rule: log_b(x^r) = r * log_b(x) - The change of base formula: log_b(x) = log_c(x) / log_c(b), where c is any positive number different from 1. These properties allow us to simplify complex logarithmic expressions and perform calculations more easily.

Ans. Logarithms have numerous real-life applications. Some examples include: - Financial calculations: Logarithms are used in finance to calculate compound interest, investment returns, and mortgage payments. - pH scale: The pH scale measures acidity and alkalinity using logarithms. Each unit on the pH scale represents a tenfold difference in acidity or alkalinity. - Sound and decibels: The decibel scale, used to measure sound intensity, is logarithmic. Each increase of 10 decibels corresponds to a tenfold increase in sound intensity. - Earthquakes: The Richter scale, used to measure the magnitude of earthquakes, is also logarithmic. Each increase of 1 on the Richter scale represents a tenfold increase in the amplitude of seismic waves.

Ans. Logarithms are frequently used to solve exponential equations. The basic approach is to take the logarithm of both sides of the equation, using the same base. This allows us to convert the exponential equation into a simpler logarithmic equation. By applying the properties of logarithms, we can then solve for the unknown variable. Logarithms help in finding the value of the exponent that makes the equation true.

Ans. Logarithms and exponential growth are closely related. Logarithms allow us to solve equations involving exponential growth or decay. Exponential growth can be expressed as an equation of the form y = ab^x, where a is the initial value, b is the growth factor, and x is the time or number of periods. Taking the logarithm of both sides of this equation can help us determine the growth rate or find the time required to reach a certain value. Logarithms provide a useful tool for analyzing and predicting exponential growth in various fields such as population growth, compound interest, and the spread of diseases.