Solving & Graphing Exponential Equations

An exponential equation is an equation in which a variable appears in the exponent. For example, $$2^x = 8$$ .

Set the exponents equal: -2n = 6. Solve for n: n = -3.

The range is (-8, ∞). The graph approaches y = -8 but never touches it.

8 can be rewritten as $$2^3$$ .

An asymptote is a line that a graph approaches but never touches. For exponential functions, the horizontal asymptote is often y = 0.

The base is the constant that is raised to a power. In $$f(x) = a^x$$ , 'a' is the base.

Start by plotting points for various x values, then draw a smooth curve that approaches the asymptote.

To solve, rewrite 8 as $$2^3$$ , then set the exponents equal: 3 = 5x + 7. Solve for x: x = -⅘.

An exponential function is increasing if its base is greater than 1, meaning as x increases, f(x) also increases.

An exponential function is decreasing if its base is between 0 and 1, meaning as x increases, f(x) decreases.