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Let's solve each given equation step-by-step to find the solution for [tex]\( x \)[/tex]. We will then identify which equation has the same solution as the original equation [tex]\( 5^{(2x-5)} = 125 \)[/tex].
### Original Equation
1. [tex]\(5^{(2x-5)} = 125\)[/tex]
2. Convert [tex]\( 125 \)[/tex] to a base of 5: [tex]\( 125 = 5^3 \)[/tex]
3. Hence, [tex]\(5^{(2x-5)} = 5^3\)[/tex]
4. Since the bases are the same, the exponents must be equal: [tex]\( 2x - 5 = 3 \)[/tex]
5. Solve for [tex]\( x \)[/tex]:
[tex]\[ 2x - 5 = 3 \\ 2x = 3 + 5 \\ 2x = 8 \\ x = 4 \][/tex]
The solution for the original equation is [tex]\( x = 4 \)[/tex].
### Check Each Given Equation
#### Equation 1: [tex]\(64 = 2^{(3x-6)}\)[/tex]
1. Convert [tex]\( 64 \)[/tex] to a base of 2: [tex]\( 64 = 2^6 \)[/tex]
2. Hence, [tex]\(2^{(3x-6)} = 2^6\)[/tex]
3. Since the bases are the same, the exponents must be equal: [tex]\( 3x - 6 = 6 \)[/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[ 3x - 6 = 6 \\ 3x = 6 + 6 \\ 3x = 12 \\ x = 4 \][/tex]
The solution for Equation 1 is [tex]\( x = 4 \)[/tex].
#### Equation 2: [tex]\(343 = 7^{(2x+9)}\)[/tex]
1. Convert [tex]\( 343 \)[/tex] to a base of 7: [tex]\( 343 = 7^3 \)[/tex]
2. Hence, [tex]\(7^{(2x+9)} = 7^3\)[/tex]
3. Since the bases are the same, the exponents must be equal: [tex]\( 2x + 9 = 3 \)[/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[ 2x + 9 = 3 \\ 2x = 3 - 9 \\ 2x = -6 \\ x = -3 \][/tex]
The solution for Equation 2 is [tex]\( x = -3 \)[/tex].
#### Equation 3: [tex]\(3^{(5x-1)} = 81\)[/tex]
1. Convert [tex]\( 81 \)[/tex] to a base of 3: [tex]\( 81 = 3^4 \)[/tex]
2. Hence, [tex]\(3^{(5x-1)} = 3^4\)[/tex]
3. Since the bases are the same, the exponents must be equal: [tex]\( 5x - 1 = 4 \)[/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[ 5x - 1 = 4 \\ 5x = 4 + 1 \\ 5x = 5 \\ x = 1 \][/tex]
The solution for Equation 3 is [tex]\( x = 1 \)[/tex].
#### Equation 4: [tex]\(4^{(7x+11)} = 256\)[/tex]
1. Convert [tex]\( 256 \)[/tex] to a base of 4: [tex]\( 256 = 4^4 \)[/tex]
2. Hence, [tex]\(4^{(7x+11)} = 4^4\)[/tex]
3. Since the bases are the same, the exponents must be equal: [tex]\( 7x + 11 = 4 \)[/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[ 7x + 11 = 4 \\ 7x = 4 - 11 \\ 7x = -7 \\ x = -1 \][/tex]
The solution for Equation 4 is [tex]\( x = -1 \)[/tex].
### Conclusion
The original equation [tex]\( 5^{(2x-5)} = 125 \)[/tex] has the solution [tex]\( x = 4 \)[/tex]. The equation [tex]\( 64 = 2^{(3x-6)} \)[/tex] also has the solution [tex]\( x = 4 \)[/tex]. Therefore, the correct equation that has the same solution as the original equation is:
[tex]\[ 64 = 2^{(3x-6)} \][/tex]
### Original Equation
1. [tex]\(5^{(2x-5)} = 125\)[/tex]
2. Convert [tex]\( 125 \)[/tex] to a base of 5: [tex]\( 125 = 5^3 \)[/tex]
3. Hence, [tex]\(5^{(2x-5)} = 5^3\)[/tex]
4. Since the bases are the same, the exponents must be equal: [tex]\( 2x - 5 = 3 \)[/tex]
5. Solve for [tex]\( x \)[/tex]:
[tex]\[ 2x - 5 = 3 \\ 2x = 3 + 5 \\ 2x = 8 \\ x = 4 \][/tex]
The solution for the original equation is [tex]\( x = 4 \)[/tex].
### Check Each Given Equation
#### Equation 1: [tex]\(64 = 2^{(3x-6)}\)[/tex]
1. Convert [tex]\( 64 \)[/tex] to a base of 2: [tex]\( 64 = 2^6 \)[/tex]
2. Hence, [tex]\(2^{(3x-6)} = 2^6\)[/tex]
3. Since the bases are the same, the exponents must be equal: [tex]\( 3x - 6 = 6 \)[/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[ 3x - 6 = 6 \\ 3x = 6 + 6 \\ 3x = 12 \\ x = 4 \][/tex]
The solution for Equation 1 is [tex]\( x = 4 \)[/tex].
#### Equation 2: [tex]\(343 = 7^{(2x+9)}\)[/tex]
1. Convert [tex]\( 343 \)[/tex] to a base of 7: [tex]\( 343 = 7^3 \)[/tex]
2. Hence, [tex]\(7^{(2x+9)} = 7^3\)[/tex]
3. Since the bases are the same, the exponents must be equal: [tex]\( 2x + 9 = 3 \)[/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[ 2x + 9 = 3 \\ 2x = 3 - 9 \\ 2x = -6 \\ x = -3 \][/tex]
The solution for Equation 2 is [tex]\( x = -3 \)[/tex].
#### Equation 3: [tex]\(3^{(5x-1)} = 81\)[/tex]
1. Convert [tex]\( 81 \)[/tex] to a base of 3: [tex]\( 81 = 3^4 \)[/tex]
2. Hence, [tex]\(3^{(5x-1)} = 3^4\)[/tex]
3. Since the bases are the same, the exponents must be equal: [tex]\( 5x - 1 = 4 \)[/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[ 5x - 1 = 4 \\ 5x = 4 + 1 \\ 5x = 5 \\ x = 1 \][/tex]
The solution for Equation 3 is [tex]\( x = 1 \)[/tex].
#### Equation 4: [tex]\(4^{(7x+11)} = 256\)[/tex]
1. Convert [tex]\( 256 \)[/tex] to a base of 4: [tex]\( 256 = 4^4 \)[/tex]
2. Hence, [tex]\(4^{(7x+11)} = 4^4\)[/tex]
3. Since the bases are the same, the exponents must be equal: [tex]\( 7x + 11 = 4 \)[/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[ 7x + 11 = 4 \\ 7x = 4 - 11 \\ 7x = -7 \\ x = -1 \][/tex]
The solution for Equation 4 is [tex]\( x = -1 \)[/tex].
### Conclusion
The original equation [tex]\( 5^{(2x-5)} = 125 \)[/tex] has the solution [tex]\( x = 4 \)[/tex]. The equation [tex]\( 64 = 2^{(3x-6)} \)[/tex] also has the solution [tex]\( x = 4 \)[/tex]. Therefore, the correct equation that has the same solution as the original equation is:
[tex]\[ 64 = 2^{(3x-6)} \][/tex]
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