Functions - Exponential and Logarithmic Functions

The Exponential Function $f(x) = a^x$: An exponential function is defined by a constant base $a > 0$ and $a \neq 1$. Visually, the graph is a smooth curve that always stays above the x-axis, meaning the x-axis ($y = 0$) serves as a horizontal asymptote. If $a > 1$, the graph rises steeply to the right (exponential growth), whereas if $0 < a < 1$, it falls toward the x-axis (exponential decay). All such functions pass through the y-intercept $(0, 1)$.

The Natural Exponential Function $f(x) = e^x$: The constant $e \approx 2.71828$ is the unique base for which the slope of the tangent line to the curve at the point $(0, 1)$ is exactly $1$. Its graph is a specific case of exponential growth and is the primary tool for modeling continuous growth or decay processes in science and finance.

Logarithmic Functions as Inverses: The function $f(x) = \log_a(x)$ is defined as the inverse of the exponential function $f(x) = a^x$. Geometrically, this means the graph of a logarithm is the reflection of the exponential graph across the diagonal line $y = x$. Unlike exponential functions, logarithmic functions have a vertical asymptote at $x = 0$ (the y-axis) and pass through the x-intercept $(1, 0)$. The domain is restricted to $(0, \infty)$.

Laws of Logarithms: These properties allow for the simplification of complex logarithmic and exponential expressions. The Product Law $\log_b(xy) = \log_b x + \log_b y$ turns multiplication into addition; the Quotient Law $\log_b(\frac{x}{y}) = \log_b x - \log_b y$ turns division into subtraction; and the Power Law $\log_b(x^k) = k \log_b x$ allows exponents to be brought down as coefficients.

The Natural Logarithm: The natural logarithm, written as $\ln(x)$, is simply a logarithm with base $e$. It is the most common logarithm used in higher-level mathematics. An important visual and algebraic property is that $e^{\ln x} = x$ and $\ln(e^x) = x$, demonstrating that they 'undo' each other.

Transformations of Graphs: For the general form $y = p \cdot a^{x-h} + k$ or $y = p \cdot \log_a(x-h) + k$, the constant $k$ shifts the graph vertically and moves the horizontal asymptote ($y=k$), while $h$ shifts the graph horizontally and moves the vertical asymptote ($x=h$). The multiplier $p$ causes a vertical stretch or compression and can reflect the graph across the x-axis if $p < 0$.

Change of Base Formula: This concept allows us to convert a logarithm with any base into a ratio of logarithms with a common base (usually base $e$ or base $10$). This is represented by $\log_a b = \frac{\log_c b}{\log_c a}$. Visually, changing the base of a logarithmic function results in a vertical stretch or compression of the curve.