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Solve the exponential equation for x. {:[3^(3x-2)=9^(4x-1)],[x=◻]:}

Solve the exponential equation for x x .\newline33x−2=94x−1x=□ \begin{array}{l} 3^{3 x-2}=9^{4 x-1} \\ x=\square \end{array}

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Full solution

Q. Solve the exponential equation for x x .\newline33x−2=94x−1x=□ \begin{array}{l} 3^{3 x-2}=9^{4 x-1} \\ x=\square \end{array}
  1. Recognize Power of 33: Recognize that 99 is a power of 33, specifically 9=329 = 3^2.\newlineRewrite the equation using this relationship to have the same base on both sides.\newline33x−2=(32)4x−13^{3x-2} = (3^2)^{4x-1}
  2. Rewrite Equation with Same Base: Apply the power of a power rule to the right side of the equation.\newline(32)(4x−1)=3(2⋅(4x−1))(3^2)^{(4x-1)} = 3^{(2\cdot(4x-1))}
  3. Apply Power of a Power Rule: Set the exponents equal to each other since the bases are now the same.\newline3x−2=2×(4x−1)3x - 2 = 2\times(4x - 1)
  4. Set Exponents Equal: Distribute the 22 on the right side of the equation.3x−2=8x−23x - 2 = 8x - 2

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