To determine which equations can be rewritten to have common bases on both sides, we need to express both sides of each equation using the same base. Let's analyze each of the answer choices.
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\(3^x = 4^{x+1}\)
- Left side: Base 3
- Right side: Base 4
- Cannot be rewritten to have common bases.
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\(3^x = 9^{2x-1}\)
- Left side: \(3^x\)
- Right side: \(9 = 3^2 \implies 9^{2x-1} = (3^2)^{2x-1} = 3^{4x-2}\)
- Both sides can be rewritten to base 3: \[ 3^x = 3^{4x-2} , \text{(common base: 3)} \]
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\(4^{3x-1} = 64^x\)
- Left side: \(4 = 2^2 \implies 4^{3x-1} = (2^2)^{3x-1} = 2^{6x-2}\)
- Right side: \(64 = 4^3 = (2^2)^3 = 2^6 \implies 64^x = (2^6)^x = 2^{6x}\)
- Both sides can be rewritten to base 2: \[ 2^{6x-2} = 2^{6x} , \text{(common base: 2)} \]
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\(81 = 9^x\)
- Left side: \(81 = 9^2\)
- Right side: \(9 = 3^2 \implies 9^x = (3^2)^x = 3^{2x}\)
- Rewriting both sides: \[ 9^2 = 3^{2x} \implies 3^{4} = 3^{2x} , \text{(common base: 3)} \]
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\(2^x = 6\)
- Left side: Base 2
- Right side: Base 6
- Cannot have common bases.
From the analysis, the equations that can be rewritten to have common bases on both sides are:
- \(3^x = 9^{2x-1}\)
- \(4^{3x-1} = 64^x\)
- \(81 = 9^x\)
Thus, the corresponding options are the second, third, and fourth choices.
1 answer