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Solve the equation 9^(5-x)=1/(sqrt(15^(4x)). Express your answer in terms of ln3 and ln5.

 Nov 28, 2019
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Solve the equation \(9^{5-x}= \dfrac{1}{\sqrt{15^{4x}}}\).

Express your answer in terms of \(\ln3\) and \(\ln5\) .

 

\(\begin{array}{|rcll|} \hline 9^{5-x} &=& \dfrac{1}{\sqrt{15^{4x}}} \\ 9^{5-x} &=& \dfrac{1}{15^{\frac{4x}{2}}} \\ 9^{5-x} &=& \dfrac{1}{15^{2x}} \\ 9^59^{-x} &=& \dfrac{1}{15^{2x}} \\ 9^53^{-2x} &=& \dfrac{1}{15^{2x}} \\ 9^5 &=& \dfrac{3^{2x}}{15^{2x}} \\ 9^5 &=& \left(\dfrac{3}{15}\right)^{2x} \\ 9^5 &=& \left(\dfrac{1}{5}\right)^{2x} \\ 9^5 &=& \dfrac{1}{5^{2x}} \\ 9^55^{2x} &=& 1 \\ 3^{2\cdot 5}5^{2x} &=& 1 \\ 3^{10}5^{2x} &=& 1 \\ 5^{2x} &=& 3^{-10} \quad | \quad \ln \text{ both sides } \\ 2x\ln(5) &=& -10\ln(3) \quad | \quad : 2 \\ x\ln(5) &=& -5\ln(3) \quad | \quad : 2 \\\\ \mathbf{x} &=& \mathbf{ -\dfrac{5\ln(3)}{\ln(5)} } \\ \hline \end{array}\)

 

laugh

 Nov 29, 2019

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