Asked by h
Solve the following equation. 3^x−7=27^2x
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Answered by
GPT-5 mini
AI
There are two ways to read the expression. I’ll solve both.
1) If the equation is 3^{x-7} = 27^{2x}:
Write 27 = 3^3, so 3^{x-7} = (3^3)^{2x} = 3^{6x}. Equate exponents:
x − 7 = 6x ⇒ −7 = 5x ⇒ x = −7/5.
2) If the equation is 3^x − 7 = 27^{2x}:
Let y = 3^x (>0). Then the equation becomes y − 7 = y^6, i.e. y^6 − y + 7 = 0. For y>0 we have y^6 − y + 7 > 0 (for 0<y<1, 7−y>6; for y≥1, y^6≥y so y^6−y+7≥7), so no real y (hence no real x) satisfy it.
So the likely intended solution is x = −7/5.
1) If the equation is 3^{x-7} = 27^{2x}:
Write 27 = 3^3, so 3^{x-7} = (3^3)^{2x} = 3^{6x}. Equate exponents:
x − 7 = 6x ⇒ −7 = 5x ⇒ x = −7/5.
2) If the equation is 3^x − 7 = 27^{2x}:
Let y = 3^x (>0). Then the equation becomes y − 7 = y^6, i.e. y^6 − y + 7 = 0. For y>0 we have y^6 − y + 7 > 0 (for 0<y<1, 7−y>6; for y≥1, y^6≥y so y^6−y+7≥7), so no real y (hence no real x) satisfy it.
So the likely intended solution is x = −7/5.
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