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Solve the exponential equation for x. {:[3^(x-9)=81^(5x+1)],[x=◻]:}

Solve the exponential equation for x x .\newline3x−9=815x+1x=□ \begin{array}{l} 3^{x-9}=81^{5 x+1} \\ x=\square \end{array}

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Full solution

Q. Solve the exponential equation for x x .\newline3x−9=815x+1x=□ \begin{array}{l} 3^{x-9}=81^{5 x+1} \\ x=\square \end{array}
  1. Recognize Power of 33: Recognize that 8181 is a power of 33, since 81=3481 = 3^4.
  2. Rewrite Using Exponent Rule: Rewrite the equation using the fact that 81=3481 = 3^4.\newline3(x−9)=(34)(5x+1)3^{(x-9)} = (3^4)^{(5x+1)}
  3. Apply Power of a Power Rule: Apply the power of a power rule, which states that (ab)c=ab∗c(a^b)^c = a^{b*c}.3(x−9)=34∗(5x+1)3^{(x-9)} = 3^{4*(5x+1)}
  4. Set Exponents Equal: Since the bases are the same, we can set the exponents equal to each other. x−9=4∗(5x+1)x - 9 = 4*(5x + 1)
  5. Distribute and Simplify: Distribute the 44 on the right side of the equation.x−9=20x+4x - 9 = 20x + 4
  6. Combine Like Terms: Move all terms involving xx to one side of the equation.x−20x=4+9x - 20x = 4 + 9
  7. Divide to Solve: Combine like terms.\newline−19x=13-19x = 13
  8. Final Answer: Divide both sides by −19-19 to solve for xx.x=13(−19)x = \frac{13}{(-19)}
  9. Final Answer: Divide both sides by −19-19 to solve for xx.x=13(−19)x = \frac{13}{(-19)}Simplify the fraction to get the final answer.x=−1319x = -\frac{13}{19}

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