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Solve the exponential equation for x. {:[5^(7x+5)=125^(2x-7)],[x=◻]:}

Solve the exponential equation for x x .\newline57x+5=1252x−7x=□ \begin{array}{l} 5^{7 x+5}=125^{2 x-7} \\ x=\square \end{array}

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Q. Solve the exponential equation for x x .\newline57x+5=1252x−7x=□ \begin{array}{l} 5^{7 x+5}=125^{2 x-7} \\ x=\square \end{array}
  1. Recognize Power of 55: First, recognize that 125125 is a power of 55, specifically 125=53125 = 5^3. This will allow us to rewrite the equation with a common base.\newline57x+5=1252x−75^{7x+5} = 125^{2x-7}\newline57x+5=(53)2x−75^{7x+5} = (5^3)^{2x-7}
  2. Apply Power Rule: Next, apply the power of a power rule to simplify the right side of the equation.\newline57x+5=53(2x−7)5^{7x+5} = 5^{3(2x-7)}
  3. Set Exponents Equal: Since the bases are now the same, we can set the exponents equal to each other. 7x+5=3(2x−7)7x + 5 = 3(2x - 7)
  4. Distribute and Simplify: Distribute the 33 on the right side of the equation.7x+5=6x−217x + 5 = 6x - 21
  5. Isolate Variable x: Now, isolate the variable x by moving terms involving x to one side and constants to the other. \newline7x−6x+5=6x−6x−217x - 6x + 5 = 6x - 6x - 21\newlinex+5=−21x + 5 = -21
  6. Subtract and Solve for x: Subtract 55 from both sides to solve for x.\newlinex+5−5=−21−5x + 5 - 5 = -21 - 5\newlinex=−26x = -26

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