Logarithmic function

Definition

A logarithmic function is the inverse of an exponential function, typically written as $f(x) = \log_b(x)$ where $b$ is the base. Common bases include base 10 (common logarithm) and base e (natural logarithm).

5 Must Know Facts For Your Next Test

1. The natural logarithm, denoted as $\ln(x)$, has a base of $e \approx 2.71828$.
2. The derivative of $\ln(x)$ is $\frac{1}{x}$.
3. The integral of $\ln(x)$ can be found using integration by parts: $$\int \ln(x) \, dx = x \ln(x) - x + C$$.
4. Logarithmic functions are useful in solving integrals involving products of polynomials and exponentials.
5. Properties such as $\log_b(xy) = \log_b(x) + \log_b(y)$ and $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$ simplify complex expressions.

Review Questions

• What is the integral of $\ln(x)$?
• What are the properties of logarithms that can simplify integration?
• How do you express the natural logarithm in terms of its derivative?

Related terms

Exponential Function: A function of the form $f(x) = b^x$ where $b > 0$ and $b \neq 1$. It is the inverse function of a logarithmic function.

Integration by Parts:

A technique for integrating products of functions, based on the formula: $$\int u \, dv = uv - \int v \, du$$.

$e$ (Euler's Number): An irrational constant approximately equal to 2.71828, which serves as the base for natural logarithms.