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Solve the exponential equation for x. {:[7^(3-5x)=((1)/(49))^(2x+9)],[x=◻]:}

Solve the exponential equation for x x .\newline73−5x=(149)2x+9x=□ \begin{array}{l} 7^{3-5 x}=\left(\frac{1}{49}\right)^{2 x+9} \\ x=\square \end{array}

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Q. Solve the exponential equation for x x .\newline73−5x=(149)2x+9x=□ \begin{array}{l} 7^{3-5 x}=\left(\frac{1}{49}\right)^{2 x+9} \\ x=\square \end{array}
  1. Recognize Common Base: Recognize that 73−5x7^{3-5x} and (149)2x+9\left(\frac{1}{49}\right)^{2x+9} can be written with a common base.\newlineSince 4949 is 727^2, we can rewrite the equation as:\newline73−5x=(7−2)2x+97^{3-5x} = \left(7^{-2}\right)^{2x+9}
  2. Apply Power of a Power Rule: Apply the power of a power rule to the right side of the equation.\newlineThe power of a power rule states that (am)n=amn(a^m)^n = a^{mn}. Therefore:\newline73−5x=7−4x−187^{3-5x} = 7^{-4x-18}
  3. Set Exponents Equal: Since the bases are the same, we can set the exponents equal to each other.\newline3−5x=−4x−183 - 5x = -4x - 18
  4. Solve for x: Solve for x by isolating the variable.\newlineAdd 5x5x to both sides:\newline3=x−183 = x - 18\newlineAdd 1818 to both sides:\newline21=x21 = x

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