Concepts to remember on Logarithm while preparing for JEE exam and CBSE board exams:

• **Definition of logarithm**: The logarithm of a number $x$ to the base $b$, denoted as ${\log }_{b}x$, is the exponent to which $b$ must be raised to obtain $x$.
• **Laws of logarithms**: - ${\log }_b(xy)={\log }_{b}x+{\log }_{b}y$ -${\log }_b\left(\frac{x}{y}\right)={\log }_{b}x-{\log }_{b}y$ - ${\log }_b{x}^n=n{\log }_{b}x$ -${\log }_{b}b=1$ -${\log }_{b}1=0$
• **Common logarithms and natural logarithms**: Common logarithms have a base of 10, and are denoted as $\log x$. Natural logarithms have a base of $e$ and are denoted as $\ln x$.
• **Change of base of logarithms**: The logarithm of a number $x$ to the base $b$ can be expressed in terms of its logarithm to any other base $a$ using the formula: ${\log }_{b}x=\frac{{\log }_{a}x}{{\log }_{a}b}$.
• **Logarithmic functions and their graphs**: The logarithmic function is a function of the form $f(x)={\log }_{b}x$, where $b>0$ and $b\ne 1$. The graph of a logarithmic function is an increasing curve that passes through the point $(1,0)$.
• **Solution of logarithmic equations**: Logarithmic equations can be solved by rewriting them in exponential form and then simplifying. For example, the equation ${\log }_{2}x=3$ can be rewritten as $2^3=x$, which simplifies to $x=8$.
• **Applications of logarithms**: Logarithms have many applications in various fields, including: - pH calculations: The pH of a solution is defined as $-log[H^{+}]$, where $[H^{+}]$ is the concentration of hydrogen ions in moles per liter. - Sound intensity: The intensity of a sound wave is measured in decibels (dB), which is defined as $10\log \left(\frac{I}{I_0}\right)$, where $I$ is the intensity of the sound wave and $I_0$ is a reference intensity. - Electrical engineering: Logarithms are used in the design and analysis of electrical circuits, such as amplifiers and filters.
• **Inverse trigonometric functions**: Inverse trigonometric functions are functions that undo the trigonometric functions. The inverse trigonometric functions are: - $si{n}^{-1}x$, which is the inverse of the sine function - $co{s}^{-1}x$, which is the inverse of the cosine function - $ta{n}^{-1}x$, which is the inverse of the tangent function
• **Properties of inverse trigonometric functions**: - The inverse trigonometric functions are one-to-one functions. - The inverse trigonometric functions are increasing functions. - The inverse trigonometric functions have a restricted domain and range.
• **Graphs of inverse trigonometric functions**: The graphs of the inverse trigonometric functions are: - The graph of $si{n}^{-1}x$ is a curve that passes through the point $(0,0)$ and has a range of $-\frac{\pi }{2}$ to $\frac{\pi }{2}$. - The graph of $co{s}^{-1}x$ is a curve that passes through the point $(1,0)$ and has a range of $0$ to $\pi$. - The graph of $ta{n}^{-1}x$ is a curve that passes through the point $(0,0)$ and has a range of $-\frac{\pi }{2}$ to $\frac{\pi }{2}$.
• **Solution of inverse trigonometric equations**: Inverse trigonometric equations can be solved by using the inverse trigonometric functions. For example, the equation $si{n}^{-1}x=30\text{degree}$ can be solved by finding the angle whose sine is $30\text{degree}$. This angle is found to be $\frac{\pi }{6}$ radians.
• **Applications of inverse trigonometric functions**: Inverse trigonometric functions have many applications in various fields, including: - Surveying: Inverse trigonometric functions are used to find the angles between two points when the distance between the points and the angle between the line connecting the points and a reference line are known. - Navigation: Inverse trigonometric functions are used to find the course of a ship when the ship's position and the direction of the destination are known. - Robotics: Inverse trigonometric functions are used to control the movement of robots by calculating the angles at which the robot's joints should be rotated.