Sure! Let's solve each exponential equation step by step to find the correct answers.
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Equation: \(5^{27} = 5^{-9x}\)
Since the bases are the same, we can equate the exponents: \[ 27 = -9x \] To solve for \(x\), divide both sides by \(-9\): \[ x = \frac{27}{-9} = -3 \]
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Equation: \(3^{x-4} + 2x = 3^{3}(2x + 7)\)
Start by simplifying the right side: \[ 3^3 = 27 \] Therefore: \[ 27(2x + 7) = 54x + 189 \] Now, rewrite the equation: \[ 3^{x-4} + 2x = 54x + 189 \] Rearranging gives: \[ 3^{x-4} = 52x + 189 \] This equation may need numerical or graphical methods to solve, as it is not straightforward to isolate \(x\).
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Equation: \(12^{x-5} = 12^{3x}\)
Again, since the bases are the same, we equate the exponents: \[ x - 5 = 3x \] Rearranging gives: \[ -5 = 2x \] Thus: \[ x = -\frac{5}{2} = -2.5 \]
So far, we have the following solutions:
- For \(5^{27} = 5^{-9x}\), the solution is \(x = -3\).
- The equation \(3^{x-4} + 2x = 3^{3}(2x + 7)\) needs further numerical or graphical exploration or a specific value to be solved correctly.
- For \(12^{x-5} = 12^{3x}\), the solution is \(x = -2.5\).
If you provide specific options to match, we can proceed to identify which match these calculations!
1 answer