### Logarithm

**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**: - \(\log_b(xy)=\log_b x + \log_b y\) -\(\log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y\) - \(\log_b x^n=n\log_b x\) -\(\log_bb=1\) -\(\log_b 1=0\)
- **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: - \(sin^{-1} x\), which is the inverse of the sine function - \(cos^{-1} x\), which is the inverse of the cosine function - \(tan^{-1} x\), 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 \(sin^{-1} x\) 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 \(cos^{-1} x\) is a curve that passes through the point \((1,0)\) and has a range of \(0\) to \(\pi\). - The graph of \(tan^{-1} x\) 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 \(sin^{-1} x = 30\degree\) can be solved by finding the angle whose sine is \(30\degree\). 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.