7-6 Solving Equations Involving Exponents

Property

An exponential equation is one in which the variable is part of an exponent. Many exponential equations can be solved by writing both sides of the equation as powers with the same base. In general, if two equivalent powers have the same base, then their exponents must be equal. If $b^m = b^n$, then $m=n$ (for $b > 0, b \neq 1$).

Examples

• To solve the equation $3^x = 81$, rewrite it as $3^x = 3^4$. Equating the exponents gives the solution $x=4$.

• For the equation $5^{x-1} = 125$, write it as $5^{x-1} = 5^3$. Then, solve the simpler equation $x-1 = 3$ to get $x=4$.

Property

One-to-One Property of Exponential Equations
For $a > 0$ and $a \neq 1$,

To solve an exponential equation:

1. Write both sides of the equation with the same base, if possible.
2. Write a new equation by setting the exponents equal.
3. Solve the new equation.
4. Check the solution.

Examples

• To solve $4^{x+2} = 64$, first rewrite $64$ as $4^3$. The equation becomes $4^{x+2} = 4^3$. Now, set the exponents equal: $x+2 = 3$, which gives $x=1$.
• Solve $e^{x^2-3} = e^{2x}$. Since the bases are both $e$, we set the exponents equal: $x^2-3 = 2x$. This gives a quadratic equation $x^2-2x-3=0$, which factors to $(x-3)(x+1)=0$. So, $x=3$ or $x=-1$.
• Solve $9^{x} = 27$. Write both sides with base 3: $(3^2)^x = 3^3$, which simplifies to $3^{2x} = 3^3$. Therefore, $2x=3$, and $x=\frac{3}{2}$.

Explanation

This property is a powerful shortcut for solving exponential equations. If you can make the bases on both sides of the equation the same, you can ignore the bases and simply set the exponents equal to each other to solve.