Exponential equation
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1) What is an exponential equation? 2) Connection with exponential functions 3) Solution procedure 4) Examples of exponential equations in basic form: Example 1: Solve the equation for x ∈ ℝ: 81^x = 9 Example 2: Solve the equation for x ∈ ℝ: 3^(3x - 2) = 1 Example 3: Solve the equation for x ∈ ℝ: e^x = 1 Example 4: Solve the equation for x ∈ ℝ: 2^(x^2 - 6x - 5/2) = 16√2 Example 5: Solve the equation for x ∈ ℝ: (5/8)^[(2x + 1) / (x - 1)] = (512 / 125)^(3 - x) Example 6: Solve the equation for x ∈ ℝ: 6^(x + 2) = -6 5) Exponential equation solution using deduction: Example 7: Solve the equation for x ∈ ℝ: 2^x + 2^(x + 1) = 24 Example 8: Solve the equation for x ∈ ℝ: 9^(x + 2) + 5·9^(x + 1) = 14 6) Exponential equations leading to other interesting types of equations: Example 9: Solve the equation for x ∈ ℝ: 9^3x - 1 = 3^(8x - 2) Example 10: Solve the equation for x ∈ ℝ: 4^√(x + 1) = 64·2^√(x + 1) Example 11: Solve the equation for x ∈ ℝ: 3^(2x + 5) = 0 7) Exponential equation solution by substitution: Example 12: Solve the equation for x ∈ ℝ: 4^x + 2^x - 6 = 0 Example 13: Solve the equation for x ∈ ℝ: 3^(x + 1) + 9^x = 108 Example 14: Solve the equation for x ∈ ℝ: 2^x = 7 + 8·2^(-x) Example 15: Solve the equation for x ∈ ℝ: 2^3 · 4^x + 1 = 3^2 · 2^x 8) Summary 9) What happens next? we still lack exponential equations in which we need to take logarithms – we will return to these after logarithms
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