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Solve the exponential equation for x. {:[5^(x-12)=((1)/(125))^((12)/(5))],[x=◻]:}

Solve the exponential equation for x x .\newline5x−12=(1125)125x=□ \begin{array}{l} 5^{x-12}=\left(\frac{1}{125}\right)^{\frac{12}{5}} \\ x=\square \end{array}

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Q. Solve the exponential equation for x x .\newline5x−12=(1125)125x=□ \begin{array}{l} 5^{x-12}=\left(\frac{1}{125}\right)^{\frac{12}{5}} \\ x=\square \end{array}
  1. Recognize Exponential Form: Recognize that both sides of the equation are written in exponential form.\newlineWe can rewrite 1125\frac{1}{125} as 5−35^{-3} since 125125 is 535^3.
  2. Rewrite Using Property of Exponents: Rewrite the right side of the equation using the property of exponents.\newline(1125)125(\frac{1}{125})^{\frac{12}{5}} can be rewritten as (5−3)125(5^{-3})^{\frac{12}{5}}.
  3. Simplify Using Power Rule: Simplify the right side of the equation using the power of a power rule.\newline(5−3)125(5^{-3})^{\frac{12}{5}} simplifies to 5−3655^{-\frac{36}{5}} or 5−7.25^{-7.2}.
  4. Set Exponents Equal: Set the exponents equal to each other since the bases are the same. x−12=−7.2x - 12 = -7.2
  5. Solve for x: Solve for x by adding 1212 to both sides of the equation.\newlinex−12+12=−7.2+12x - 12 + 12 = -7.2 + 12\newlinex=4.8x = 4.8

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