To determine which of the given equations can be rewritten to have common bases on both sides, we can analyze each option for potential rewriting involving exponential expressions with common bases.
A The equation is \(3^x = 4^{x+1}\).
- Bases: 3 and 4.
- These bases do not match and cannot be rewritten as common bases.
Result: No common base.
B The equation is \(3^x = 9^{2x-1}\).
- Rewrite \(9\) as \(3^2\):
\[ 9^{2x-1} = (3^2)^{2x-1} = 3^{2(2x-1)} = 3^{4x-2} \] - Now we have \(3^x = 3^{4x-2}\), which has a common base.
Result: Common base possible.
C The equation is \(4^{3x-1} = 64^x\).
- Rewrite \(64\) as \(4^3\):
\[ 64^x = (4^3)^x = 4^{3x} \] - Now we have \(4^{3x-1} = 4^{3x}\), which has a common base.
Result: Common base possible.
D The equation is \(81 = 9^x\).
- Rewrite \(81\) as \(9^2\):
\[ 81 = 9^2 \] - The equation becomes \(9^2 = 9^x\), which has a common base.
Result: Common base possible.
E The equation is \(2^x = 6\).
- Bases: 2 and 6.
- These bases do not match and cannot be rewritten as common bases.
Result: No common base.
Summary
The equations that can be rewritten to have common bases on both sides are:
- B: \(3^x = 9^{2x-1}\)
- C: \(4^{3x-1} = 64^x\)
- D: \(81 = 9^x\)
Therefore, the correct answers are B, C, and D.
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