Logarithm

Definition

A logarithm is the inverse operation to exponentiation, meaning it undoes the process of raising a number to a power. The logarithm of a number $x$ with base $b$ is the exponent to which $b$ must be raised to produce $x$.

5 Must Know Facts For Your Next Test

1. The common logarithm has base 10 and is denoted as $\log_{10}$ or simply $\log$.
2. The natural logarithm has base $e$ (approximately 2.718) and is denoted as $\ln$.
3. Logarithmic functions are the inverses of exponential functions.
4. The change of base formula for logarithms is given by $\log_b(x) = \frac{\log_k(x)}{\log_k(b)}$, where $k$ can be any positive number.
5. Key properties include $\log_b(1) = 0$, $\log_b(b) = 1$, and the product rule: $\log_b(xy) = \log_b(x) + \log_b(y)$.

Review Questions

• What is the value of $\log_{10}(1000)$?
• How do you express $\ln(20)$ using a different base?
• State the inverse relationship between exponential and logarithmic functions.

Related terms

Exponential Function: A function in which an independent variable appears in the exponent, usually expressed as $f(x) = b^x$, where $b > 0$.

Base: In a logarithmic function, the base is the number that is raised to a power. For example, in $\log_2(8) = x$, the base is 2.

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