(5/3)^-5*(5/3)^-11=(5/3)^(8*x)

Find the value of x

asked by guest
on Nov 25, 2024 at 5:36 am



You asked:

Solve the equation \[{\left( \frac{5}{3} \right)}^{-5} {\left( \frac{5}{3} \right)}^{-11} = {\left( \frac{5}{3} \right)}^{8 x}\] for the unknown \(x\).

MathBot Answer:

The 4 solutions to the equation are: \[\begin{aligned}x &= -2\\x &= - \frac{- 2 \log{\left(5 \right)} + 2 \log{\left(3 \right)}}{- \log{\left(5 \right)} + \log{\left(3 \right)}} + \frac{i \pi}{- \log{\left(5 \right)} + \log{\left(3 \right)}} \approx -2.0 -6.1500295 i\\x &= - \frac{- 8 \log{\left(5 \right)} + 8 \log{\left(3 \right)}}{4 \left(- \log{\left(5 \right)} + \log{\left(3 \right)}\right)} + \frac{i \pi}{4 \left(- \log{\left(5 \right)} + \log{\left(3 \right)}\right)} \approx -2.0 -1.5375074 i\\x &= - \frac{- 4 \log{\left(5 \right)} + 4 \log{\left(3 \right)}}{2 \left(- \log{\left(5 \right)} + \log{\left(3 \right)}\right)} + \frac{i \pi}{2 \left(- \log{\left(5 \right)} + \log{\left(3 \right)}\right)} \approx -2.0 -3.0750147 i\end{aligned}\]

\(i\) is the imaginary unit, defined as \(i^2 = -1\).

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