Title: How to Solve Exponential and Logarithmic Equations – Step-by-Step Examples

This post provides two clear examples of solving both logarithmic and exponential equations—one with properties of logarithms and one using exponential factoring. These are essential techniques for algebra, precalculus, and even calculus.

Example 1: Solve

log₈(x + 5) – log₈(x – 2) = 1

Step-by-step:

1. Apply the log quotient rule:
   log₈((x + 5)/(x – 2)) = 1

2. Convert to exponential:
   (x + 5)/(x – 2) = 8

3. Cross-multiply and solve:
   x + 5 = 8(x – 2) → x = 3

4. Check your solution:
   log₈(8) – log₈(1) = 1 – 0 = 1 ✅
   Final answer: x = 3

Example 2: Solve

x²e²ˣ + 2xe²ˣ = 8e²ˣ

Step-by-step:

1. Factor out e²ˣ:
   e²ˣ(x² + 2x – 8) = 0

2. Solve the quadratic:
   (x – 2)(x + 4) = 0 → x = 2 or x = –4

3. Check both solutions:

   • x = 2 → ✅

   • x = –4 → ✅

4. Reminder: Never divide both sides by e²ˣ in early-level courses or exams—it can cause you to lose solutions.

Final Answer:

• Logarithmic Equation: x = 3

• Exponential Equation: x = 2, –4

These pages are pulled from Chapter 5 of the Ultimate Crash Course for STEM Majors, created to build problem-solving habits that apply in calculus, engineering, and higher math.

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