Question

Which of the following equations can be rewritten to have common bases on both sides?

A
3x=4x+13^x=4^{x+1}3
x
=4
x+1


B
3x=92x−13^x=9^{2x-1}3
x
=9
2x−1


C
43x−1=64x4^{3x-1}=64^x4
3x−1
=64
x


D
81=9x81=9^x81=9
x


E
2x=62^x=62
x
=6

Answer 1

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.