Solving Exponential Equations

Exponential equations are equations in which variables appear in the exponents. To solve these equations, we often use properties of exponents and logarithms. The goal is to isolate the variable and find its value.

Example

Solve the equation: $2^{x+1} = 16$

1. Recognize that $16$ can be rewritten as a power of $2$:
   $16 = 2^{4}$

2. Rewrite the equation:
   $2^{x+1} = 2^{4}$

3. Since the bases are the same, set the exponents equal to each other:
   $x + 1 = 4$

4. Solve for $x$:
   $x = 4 - 1$

5. Thus,
   $x = 3$

Key Questions

• What is the first step in solving an equation like $3^{x} = 81$?
• How do you handle equations where the bases are not the same?
• What role do logarithms play in solving exponential equations?
• Can you solve $5^{x} = 25$ using different methods? If so, how?

Keywords

• Exponential equations
• Base
• Exponent
• Logarithm
• Isolate