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Solve the exponential equation for x. {:[3^(4x+5)=81^(-(3)/(4))],[x=◻]:}

Solve the exponential equation for x x .\newline34x+5=81−34x=□ \begin{array}{l} 3^{4 x+5}=81^{-\frac{3}{4}} \\ x=\square \end{array}

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Full solution

Q. Solve the exponential equation for x x .\newline34x+5=81−34x=□ \begin{array}{l} 3^{4 x+5}=81^{-\frac{3}{4}} \\ x=\square \end{array}
  1. Identify base of expressions: Identify the base of the exponential expressions.\newlineThe base of the left-hand side is 33. The right-hand side can be rewritten with a base of 33 because 8181 is 343^4.
  2. Rewrite right-hand side with base 33: Rewrite the right-hand side of the equation using the base of 33.81=3481 = 3^4, so 81−(3)/(4)=(34)−(3)/(4)=3−381^{-(3)/(4)} = (3^4)^{-(3)/(4)} = 3^{-3}.
  3. Set exponents equal: Set the exponents equal to each other since the bases are the same.\newline34x+5=3−33^{4x+5} = 3^{-3}\newlineThis implies that 4x+5=−34x + 5 = -3.
  4. Solve for x: Solve for x.\newlineSubtract 55 from both sides of the equation:\newline4x+5−5=−3−54x + 5 - 5 = -3 - 5\newline4x=−84x = -8\newlineDivide both sides by 44:\newlinex=−84x = \frac{-8}{4}\newlinex=−2x = -2

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