Exponential functions | Exponentials and logarithms

The variable in the exponent

In $$x^2$$ the variable is the base; in $$2^x$$ it's the exponent. That small change makes a huge difference: $$x^2$$ at $$x = 30$$ is $$900$$ , while $2^{30}$ is over a billion.

The base can't be $$1$$ (that gives a constant) or negative (then values like $(-4)^{1/2}$ aren't real numbers).

Growth and decay

The graph

The graphs of $$y = b^x$$ and $y = \left(\frac{1}{b}\right)^x$ are mirror images across the $$y$$ -axis, because $\left(\frac{1}{b}\right)^x = b^{-x}$ .

Transformations

$y = a \cdot b^{x - h} + k$ shifts the graph $$h$$ units horizontally and $$k$$ units vertically. The asymptote moves to $$y = k$$ .

The natural base $$e$$

Compute $\left(1 + \frac{1}{n}\right)^n$ for larger and larger $$n$$ : $$2$$ , $$2.25$$ , $$2.37$$ , ..., and for $$n = 1000$$ about $$2.7169$$ . The values approach a famous irrational number:

e = 2.71828\ldots

The function $$f(x) = e^x$$ is the natural exponential function. It describes continuous growth and decay and is the most important exponential function in calculus.

Practice

Quick check

What is the $$y$$ -intercept of $f(x) = a \cdot b^x$ ?