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Solve the exponential equation for x. {:[(5^(4x+3))/(25^(9-x))=5^(2x+5)],[x=]:}

Solve the exponential equation for x x .\newline54x+3259−x=52x+5x=□ \begin{array}{l} \frac{5^{4 x+3}}{25^{9-x}}=5^{2 x+5} \\ x=\square \end{array}

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Q. Solve the exponential equation for x x .\newline54x+3259−x=52x+5x=□ \begin{array}{l} \frac{5^{4 x+3}}{25^{9-x}}=5^{2 x+5} \\ x=\square \end{array}
  1. Recognize and simplify the equation: Simplify the equation by recognizing that 2525 is a power of 55.\newlineSince 25=52 25 = 5^2 , we can rewrite 259−x 25^{9-x} as (52)9−x (5^2)^{9-x} .\newlineThis gives us 54x+3/52(9−x)=52x+5 5^{4x+3} / 5^{2(9-x)} = 5^{2x+5} .
  2. Apply properties of exponents: Apply the properties of exponents to simplify the equation further.\newlineUsing the property am−n=am/an a^{m-n} = a^m / a^n , we can combine the exponents on the left side of the equation.\newlineThis gives us 54x+3−2(9−x)=52x+5 5^{4x+3 - 2(9-x)} = 5^{2x+5} .
  3. Simplify the exponent: Simplify the exponent on the left side of the equation.\newlineCalculate 4x+3−2(9−x) 4x+3 - 2(9-x) which simplifies to 4x+3−18+2x 4x+3 - 18 + 2x .\newlineThis simplifies further to 6x−15 6x - 15 .\newlineNow we have 56x−15=52x+5 5^{6x-15} = 5^{2x+5} .
  4. Set the exponents equal: Since the bases are the same and the equation is an equality, we can set the exponents equal to each other.\newlineThis gives us 6x−15=2x+5 6x - 15 = 2x + 5 .
  5. Solve the linear equation: Solve the linear equation for x.\newlineSubtract 2x 2x from both sides to get 4x−15=5 4x - 15 = 5 .\newlineThen add 15 15 to both sides to get 4x=20 4x = 20 .\newlineFinally, divide both sides by 4 4 to find x=5 x = 5 .

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