Solving Exponential Equations

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

Rewrite 4 as $$2^2$$ , so the equation becomes $$2^x = (2^2)^{x+1}$$ . Then, set the exponents equal: $$x = 2(x + 1)$$ .

Rewrite 25 as $$5^2$$ , so the equation becomes $$5^{-3x - 1} = 5^2$$ . Then, set the exponents equal.

It means that if $$a^m = a^n$$ , then $$m = n$$ , provided that $$a > 0$$ and $$a \neq 1$$ .

36 can be rewritten as $$6^2$$ .

The solution is $$x = 3$$ , since $$8 = 2^3$$ .

The property states that $$a^m \cdot a^n = a^{m+n}$$ .

Set the exponents equal: $$2x - 3 = 6x + 5$$ .

Rewrite 125 as $$5^3$$ , so $$x + 1 = 3$$ , leading to $$x = 2$$ .

The base is the constant that is raised to a power, such as $$a$$ in $$a^x$$ .