Number and Algebra - Exponents and Logarithms

Laws of Exponents: These fundamental rules simplify operations with powers. For multiplication, $a^m \cdot a^n = a^{m+n}$, and for division, $\frac{a^m}{a^n} = a^{m-n}$. When a power is raised to another power, $(a^m)^n = a^{mn}$. Visually, an exponential growth function $f(x) = a^x$ where $a > 1$ appears as a curve that rises increasingly steeply from left to right, crossing the y-axis at $(0, 1)$ and approaching the x-axis as a horizontal asymptote ($y=0$) for negative values of $x$.

Logarithmic and Exponential Relationship: A logarithm is the inverse of an exponentiation. The statement $x = a^y$ is equivalent to $y = \log_a x$. Graphically, the logarithmic function $y = \log_a x$ is a reflection of the exponential function $y = a^x$ across the line $y = x$. While exponential graphs have horizontal asymptotes, logarithmic graphs have a vertical asymptote at $x = 0$, meaning the graph approaches but never touches the y-axis.

Laws of Logarithms: These rules allow for the expansion and condensation of logarithmic expressions. The Product Law states $\log_a (xy) = \log_a x + \log_a y$, the Quotient Law states $\log_a (\frac{x}{y}) = \log_a x - \log_a y$, and the Power Law states $\log_a (x^k) = k \log_a x$. These laws are essential for solving equations where the unknown variable is located in the exponent.

The Natural Logarithm and 'e': The irrational number $e \approx 2.718$ is the base for the natural logarithm, written as $\ln x$ (which means $\log_e x$). The function $f(x) = e^x$ is unique because its gradient at any point is equal to its y-value. In real-world contexts, $e$ is frequently used to model continuous growth or decay, such as population dynamics or radioactive half-life.

Change of Base Formula: To calculate or simplify logarithms with bases not available on a standard calculator, we use the formula $\log_a b = \frac{\log_c b}{\log_c a}$. This allows any logarithm to be converted into a base that is easier to work with, typically base 10 (common log) or base $e$ (natural log).

Negative and Rational Exponents: A negative exponent represents a reciprocal: $a^{-n} = \frac{1}{a^n}$. A rational exponent represents a root: $a^{\frac{1}{n}} = \sqrt[n]{a}$ and $a^{\frac{m}{n}} = \sqrt[n]{a^m}$. Visually, as the root $n$ increases for $x^{\frac{1}{n}}$, the curve becomes flatter for $x > 1$ while still passing through the anchor point $(1, 1)$.

Solving Exponential Equations: When solving for $x$ in equations like $a^x = b$, we apply logarithms to both sides. This transforms the exponent into a coefficient using the Power Law: $x \log a = \log b$. If the bases on both sides of an equation can be written as powers of the same number (e.g., $2^x = 8$), we equate the exponents directly ($2^x = 2^3 \implies x = 3$).