Section 6.3: Exponential Equations
This section covers solving exponential equations using properties of exponents and logarithms.
Example 1: Same Base Method
Solve \( 2^{x+1} = 16 \).
Step 1: Rewrite 16 as a power of 2 → \( 16 = 2^4 \)
Step 2: \( 2^{x+1} = 2^4 \) → x + 1 = 4
Step 3: Solve → x = 3
Example 2: Using Logarithms
Solve \( 3^{2x-1} = 20 \).
Step 1: Take log base 3: \( 2x - 1 = \log_3 20 \)
Step 2: Solve: \( 2x = \log_3 20 + 1 \)
Step 3: x = \( (\log_3 20 + 1)/2 \)
Practice Problems
- Solve \( 5^{x-2} = 125 \)
- Solve \( 2^{3x} = 32 \)
- Solve \( 4^{x+1} = 1/16 \)
- Solve \( 7^{2x} = 49 \)
- Solve \( 3^x = 5 \) (use logarithms)