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Solve the exponential equation for x. {:[27^(3x-7)=9^(6x-1)],[x=◻]:}

Solve the exponential equation for x x .\newline273x−7=96x−1x=□ \begin{array}{l} 27^{3 x-7}=9^{6 x-1} \\ x=\square \end{array}

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Full solution

Q. Solve the exponential equation for x x .\newline273x−7=96x−1x=□ \begin{array}{l} 27^{3 x-7}=9^{6 x-1} \\ x=\square \end{array}
  1. Recognize Powers of 33: Recognize that both 2727 and 99 are powers of 33. We can rewrite them as 333^3 and 323^2 respectively.\newline273x−7=(33)3x−727^{3x-7} = (3^3)^{3x-7}\newline96x−1=(32)6x−19^{6x-1} = (3^2)^{6x-1}
  2. Apply Power Rule: Apply the power of a power rule, which states that (am)n=amn(a^m)^n = a^{mn}, to both sides of the equation.\newline(33)3x−7=33(3x−7)(3^3)^{3x-7} = 3^{3(3x-7)}\newline(32)6x−1=32(6x−1)(3^2)^{6x-1} = 3^{2(6x-1)}
  3. Simplify Exponents: Simplify the exponents on both sides of the equation.\newline33(3x−7)=39x−213^{3(3x-7)} = 3^{9x-21}\newline32(6x−1)=312x−23^{2(6x-1)} = 3^{12x-2}
  4. Set Exponents Equal: Since the bases are the same, we can set the exponents equal to each other.\newline9x−21=12x−29x - 21 = 12x - 2
  5. Solve for x: Solve for x by moving the terms involving x to one side and the constant terms to the other side.\newlineSubtract 99x from both sides:\newline−21=3x−2-21 = 3x - 2
  6. Isolate x Term: Add 22 to both sides to isolate the term with x.\newline−21+2=3x−2+2-21 + 2 = 3x - 2 + 2\newline−19=3x-19 = 3x
  7. Divide to Solve x: Divide both sides by 33 to solve for x.\newline−193=3x3\frac{-19}{3} = \frac{3x}{3}\newline−193=x\frac{-19}{3} = x

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