Exponents and Logarithms
Exponents
An exponent is written as $a^n$, where $a$ is the base, and $n$ is the power (also known as index or exponent).
Exponents follow a set of algebraic rules:
| Rules of Exponents |
|---|
| $a^m \cdot a^n = a^{m+n}$ |
| $a^m \div a^n = a^{m-n}$ |
| $(a^m)^n = a^{mn}$ |
| $a^0 = 1$ |
| $a^1 = a$ |
| $a^{-m} = \frac{1}{a^m}$ |
| $a^{\frac{1}{m}} = \sqrt[m]{a}$ |
| $a^{\frac{n}{m}} = (\sqrt[m]{a})^n$ |
| $(ab)^n = a^n \cdot b^n$ |
| $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ |
Logarithms
The logarithmic function $\log_ab=x$ is the solution to the exponential equation $a^x=b$. In other words, $\log_a{b}$ gives the power to which the base $a$ needs to be raised in order to produce $b$. Therefore,
\[\boxed{a^x = b \Longleftrightarrow \log_a{b}=x}\]Base restrictions: Unless stated otherwise, $a>0, a\neq 1$, and $b>0$. (An exception would be for complex logarithms, for example.)
- The logarithm with base $e$ is written as $\ln{x}$ (natural logarithm).
- The logarithm with base $10$ is typically written as $\log{x}$ (common logarithm).
| Laws of logarithms |
|---|
| $\log_a{x} + \log_a{y} = \log_a{(xy)}$ |
| $\log_a{x} - \log_a{y} = \log_a{(\frac{x}{y})}$ |
| $n \log_a{x} = \log_a{(x^n)}$ |
Logarithmic scales
When dealing with incredibly small or large numbers, it becomes difficult to distinguish them, so they can be scaled using logarithms.
Euler’s number and the natural logarithm
Euler’s number ($e \approx 2.718\dots$) is an irrational constant, which arises in situations involving growth or decay, such as compound interest or radioactive decay.
A definition for $e$ can be found in finance, through continuous compounding:
\(e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n\) where $n$ is the number of compounding periods per unit time, and $e$ is the resulting constant as $n$ grows infinitely large (Euler’s number). See 4. Finance for more information.
The natural logarithm, written as $\ln{x}$, is the logarithm with base $e$:
\[\boxed{\ln{x} = \log_e{x}}\]Note: $e^x$ and $\ln{x}$ have unique properties in calculus. See Chapter 13. Differentiation, and Chapter 14. Integration for more information.