December 11, 2024
The logarithm function is an important function that arises from finding the inverse function of an exponential. More specifically, if \( a^y = x \), then \( y = \log_a(x) \). Definition The logarithm of \( x \) to base \( a \) is: \[\begin{align*} \log_a(x) = y \quad \text{if and only if} \quad a^y […]
November 7, 2024
The integral \( \displaystyle \int \sin x \, dx \) is a basic result in calculus, and is used in various calculations involving trigonometry, wave motion, and oscillatory behavior. Solution The antiderivative of \( \sin x \) with respect to \( x \) is: \[\begin{align*} \displaystyle \int \sin x \, dx = -\cos x + […]
November 7, 2024
The integral \( \displaystyle \int \dfrac{\ln x}{x} \, dx \) is an example of integrating compund functions, a foundational skill in calculus. Further, $\ln x$ is a function that arises in various real-world contexts, as well as in classroom and test settings. Solution The antiderivative of \( \dfrac{\ln x}{x} \) with respect to \( x […]
November 7, 2024
The antiderivative of \( \dfrac{1}{x} \) is a foundational result in calculus, and appears in a variety of contexts. Specifically, it reveals interesting properties of the natural logarithm in calculus. Solution The antiderivative of \( \dfrac{1}{x} \) with respect to \( x \) is: \[\begin{align*} \displaystyle \int \dfrac{1}{x} \, dx = \ln |x| + C […]
October 28, 2024
Multiplying exponents is simple with the \textbf{product of powers rule}: if two terms have the same base, add their exponents. Product of Powers Rule If two expressions share the same base, we add the exponents: \[\begin{align*} a^m \cdot a^n = a^{m+n} \end{align*}\] Example: \[\begin{align*} x^2 \cdot x^5 = x^{2+5} = x^7 \end{align*}\] Why It Works […]
October 28, 2024
Madhavendra Thakur October 28, 2024 Introduction Exponents allow us to represent repeated multiplication in a concise way. Mastering the basic exponent rules helps simplify complex algebraic expressions, and is a key stepping stone to more complex areas of math. This guide presents the rules with step-by-step examples for clarity. 1. Exponent Rules 1.1 Product of […]
October 28, 2024
In this blog, we’ll walk through how to evaluate the expression \( 2^5 \). Exponents are a fundamental concept in mathematics, and understanding them starts with examples like this. What Does $2^5$ Mean? The expression \( 2^5 \) means that we multiply the number 2 by itself a total of 5 times. More generally, an […]
