1. **Problem Statement:** Solve the exponential equation and convert it into an algebraic expression. 2. **General Formula:** Exponential equations often have the form $a^{f(x)} = b$, where $a$ and $b$ are constants and $f(x)$ is an algebraic expression. 3. **Important Rules:** - If $a^{m} = a^{n}$, then $m = n$ (provided $a > 0$ and $a \neq 1$). - Use logarithms to solve equations when bases differ. 4. **Example 1:** Solve $2^{x+1} = 8$ - Express 8 as a power of 2: $8 = 2^3$ - So, $2^{x+1} = 2^3$ - By the rule, $x + 1 = 3$ - Solve for $x$: $x = 3 - 1 = 2$ 5. **Example 2:** Solve $3^{2x} = 81$ - Express 81 as a power of 3: $81 = 3^4$ - So, $3^{2x} = 3^4$ - Equate exponents: $2x = 4$ - Solve for $x$: $x = 2$ 6. **Example 3:** Solve $5^{x} = 20$ - Since 20 is not a power of 5, take logarithm on both sides: $$x = \frac{\log(20)}{\log(5)}$$ - This is the algebraic expression for $x$. 7. **Summary:** Exponential equations can be converted to algebraic expressions by rewriting terms with the same base or using logarithms when bases differ. This allows solving for the variable in the exponent. **Final answers:** - Example 1: $x=2$ - Example 2: $x=2$ - Example 3: $x=\frac{\log(20)}{\log(5)}$