Pre-calculus: Solving for x in Exponential Equations

Exponential Equation with Different Bases

• Example: Solve (2^{x-5} = 3^{x+1})

Using Natural Logarithms

1. Take Natural Logarithm:

   • Apply natural log (ln) on both sides: [\ln(2^{x-5}) = \ln(3^{x+1})]

2. Apply Logarithm Properties:

   • Bring the exponent to the front: [(x-5) \ln(2) = (x+1) \ln(3)]

3. Expand and Simplify:

   • Distribute the ln:
     • (x \ln(2) - 5 \ln(2))
     • (x \ln(3) + \ln(3))

4. Rearranging the Equation:

   • Collect all x terms on one side: [x \ln(2) - x \ln(3) = 5 \ln(2) + \ln(3)]

5. Factor and Solve for x:

   • Factor out x: [x(\ln(2) - \ln(3)) = 5 \ln(2) + \ln(3)]
   • Divide both sides by ((\ln(2) - \ln(3))): [x = \frac{5 \ln(2) + \ln(3)}{\ln(2) - \ln(3)}]

6. Approximation:

   • Use a calculator to find the approximate value of x.

Alternative Method: Exponent Rules

1. Using Exponent Laws:

   • Convert equation using exponent laws:
     • (2^{x-5} = \frac{2^x}{32})
     • (3^{x+1} = 3^x \times 3)

2. Simplify and Rearrange:

   • Multiply both sides by 32:
     • (32 \cdot \frac{2^x}{32} = 3^x \times 3)
     • (2^x = 96 \cdot 3^x)

3. Divide and Rearrange:

   • Divide both sides by (3^x):
     • ((\frac{2}{3})^x = 96)

4. Apply Logarithms Again:

   • Use logarithms with base (\frac{2}{3}):
     • Take log base (\frac{2}{3}) on both sides.

5. Solve for x:

   • (x = \log_{\frac{2}{3}}(96))
   • Use the change of base formula to compute using a calculator: [x = \frac{\log_{10}(96)}{\log_{10}(\frac{2}{3})}]_

Homework and Challenge

• Find the approximation of x using both methods and verify if they match.
• Challenge: Expand the logarithm calculation from one method to the other.