Exponential Equations (no logs)

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

Since the bases are the same, set the exponents equal: $$3 = 4x + 2$$ . Solving gives $$x = \frac{1}{4}$$ .

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

Rewrite $$\frac{1}{125}$$ as $$5^{-3}$$ . Set the exponents equal: $$2x + 3 = -3$$ , leading to $$x = -3$$ .

Rewrite 4 as $$2^2$$ and 8 as $$2^3$$ . Thus, $$(2^3)^x = 2^2$$ leads to $$3x = 2$$ , so $$x = \frac{2}{3}$$ .

Rewrite 25 as $$5^2$$ . Set the exponents equal: $$-3x - 1 = 2$$ , solving gives $$x = -1$$ .

The general form is $$f(x) = a \times b^x$$ , where $$a$$ is a constant and $$b$$ is the base.

This is the product of powers property.

This is the quotient of powers property.

This is the power of a power property.