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Mastering Exponential & Logarithmic Equations: Step-by-Step Solutions

This guide covers solving exponential & logarithmic equations with variables in exponents or logarithmic expressions. Learn the base-exponent property and logarithmic equality. Practice solving equations like 52x-3 = 125 and log4x = -3. Follow detailed explanations and examples to enhance your understanding of these concepts.

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Mastering Exponential & Logarithmic Equations: Step-by-Step Solutions

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  1. Section 4.5 Solving Exponential and Logarithmic Equations

  2. Solving Exponential Equations • Equations with variables in the exponents, such as 3x = 40 and 53x = 25, are called exponential equations. Base-Exponent Property For any a > 0, a 1, ax = ay  x = y.

  3. Example • Solve:

  4. Check: 52x 3 = 125 52(3)  3? 125 53? 125 125 = 125 True The solution is 3.

  5. Another Property Property of Logarithmic Equality For any M > 0, N > 0, a > 0, and a 1, loga M = loga N  M = N.

  6. Example Solve: 2x = 50

  7. Example Solve:e0.25w = 12

  8. Solving Logarithmic Equations • Equations containing variables in logarithmic expressions, such as log2 x = 16 and log x + log (x + 4) = 1, are called logarithmic equations. • To solve logarithmic equations algebraically, we first try to obtain a single logarithmic expression on one side and then write an equivalent exponential equation.

  9. Example • Solve: log4x = 3

  10. Check: log4x = 3 The solution is

  11. Solve: Example

  12. Check Solution to Example Check: For x = 3: For x = 3: Negative numbers do not have real-number logarithms. The solution is 3.

  13. Example • Solve: The value 6 checks and is the solution.

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