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Match the exponential equation to the correct answer(3 points) 5 ^ n = 5 ^ (-9x); 12 ^ (x - 5) = 12 ^ (3x); 3 ^ (z - 4 + 2x) = 3 ^ (3(2x + 7)); x = - 25/3; x = - 5/2; x = - 3

1 answer

To solve the provided exponential equations, we will equate the exponents since the bases are the same.

  1. Equation: \( 5^n = 5^{-9x} \)

    • Since the bases are the same (5), we can set the exponents equal to each other: \[ n = -9x \] This doesn't have a specific value to match anything from the options.

  2. Equation: \( 12^{(x - 5)} = 12^{(3x)} \)

    • Similar to the first equation, we set the exponents equal: \[ x - 5 = 3x \]
    • Rearranging gives us \( -5 = 3x - x \) or \( -5 = 2x \): \[ x = -\frac{5}{2} \]
    • Match: \( x = -5/2 \)

  3. Equation: \( 3^{(z - 4 + 2x)} = 3^{(3(2x + 7))} \)

    • Set the exponents equal: \[ z - 4 + 2x = 3(2x + 7) \]
    • Expanding the right side: \[ z - 4 + 2x = 6x + 21 \]
    • Rearranging gives: \[ z - 4 = 6x - 2x + 21 \ z - 4 = 4x + 21 \ z = 4x + 25 \]
    • Since we don't have a specific z here, let’s instead focus on cases for x evident from the initial equation.
    • If \( x = -3 \): \[ z = 4(-3) + 25 = -12 + 25 = 13 \]
    • If \( x = -\frac{25}{3} \): \[ z = 4(-\frac{25}{3}) + 25 = -\frac{100}{3} + \frac{75}{3} = -\frac{25}{3} \]
    • Does not yield a solution aligning to the questions.
    • Therefore by algebraic structure and inference with >x inevitably leading toward z value observation of 4 potential we correlate if correctly through arranged solutions.

However, since \( 3^{(z - 4 + 2x)} = 3^{(3(2x + 7))} \), meaning adjusting correct z has consistency. To suppose through checking exponents can't equal each other correctly leading to maximum commonalities linking \( x = -3 \) through equations.

Thus final matches at this constituency:

Final Matching:

  1. Equation 1: Not applicable.
  2. Equation 2: \( x = -5/2 \)
  3. Equation 3: \( x = -3 \)