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 8, 2024
Introduction Logarithms are the inverse operations of exponents, allowing us to “undo” exponentiation and solve equations involving exponential terms. We will cover the main rules of logarithms, which make it easier to manipulate logarithmic expressions. Definition of Logarithms A logarithm answers the question: “To what exponent must a base be raised to get a certain […]
November 7, 2024
\[\begin{align*} \boxed{ e^{\ln x} = x } \end{align*}\] What is $e$? \( e \) is a special number in math, approximately equal to 2.718. It is also called Euler’s constant and is the base of a logarithm called the natural logarithm. You can think of it as the number that makes many things in math […]
November 7, 2024
In this article we will prove the logarithm rule: \[\begin{align*} \boxed{\log_{b}(b^k) = k} \end{align*}\] Based on how the logarithms work, $b$ must be positive and not equal to $1$, and $k$ can be any real number. The above rule is saying that if we take the logarithm of a number that is to the $k^{th}$ […]
November 7, 2024
The value of $\log_{10}(1000)$ is $\boxed{\log_{10}(1000) = 3}$. In this article, we will explore why. What does $\log_{10}(1000)$ mean? Let’s start with the expression: \[\begin{align*} \log_{10}(1000) = X \end{align*}\] This equation means that we are looking for a value of \(X\) such that: \[\begin{align*} 10^X = 1000 \end{align*}\] In other words, \(\log_{10}(1000)\) represents the power […]
October 28, 2024
\[\begin{align*} \displaystyle\boxed{\log_a b = x \text{ is equivalent to } a^x = b} \end{align*}\] What is $\log_a b$? This expression reads “log base a of b”. In mathematics, $\log_a b$ is called the logarithm of $b$ with base $a$. The logarithm $\log_a b$ answers the question: “What power do we need to raise $a$ to, in […]
October 28, 2024
$$\boxed{\log_b a = \dfrac{\log_k a}{\log_k b}}$$ The change of base formula for logarithms states that any $\log_b a$ can be expressed as $\dfrac{\log_k a}{\log_k b}$ where $k$ is any positive number. This formula is often used when calculating logarithms with the base that is inconvenient. What is a logarithm? A logarithm $\log_b a$ asks the […]
October 25, 2024
$\log_b M + \log_b N = \log_b (M \times N)$ Let’s learn how to add logarithms. Don’t worry – it’s simple and straightforward. What is a Logarithm? A logarithm answers the question: To what exponent must we raise the base to get a certain number? For example: $\log_2 8 = 3$ because $2^3 = 8$. […]
