if (8e^x-3)^3=125 then x=

asked by guest
on Jan 16, 2025 at 8:08 am



You asked:

Solve the equation \({\left( 8 {e}^{x} - 3 \right)}^{3} = 125\) for the variable \(x\).

MathBot Answer:

The complex solutions are: \[\begin{aligned}x &= \left\{2 \pi i n\; \middle|\; n \in \mathbb{Z}\right\} \approx \left\{6.2831853 n i\; \middle|\; n \in \mathbb{Z}\right\}\\x &= \left\{i \left(2 \pi n + \arctan{\left(5 \sqrt{3} \right)}\right) + \log{\left(\frac{\sqrt{19}}{8} \right)}\; \middle|\; n \in \mathbb{Z}\right\} \approx \left\{i \left(6.2831853 n + 1.4558354\right) - 0.60722205\; \middle|\; n \in \mathbb{Z}\right\}\\x &= \left\{i \left(- \arctan{\left(5 \sqrt{3} \right)} + 2 \pi n\right) + \log{\left(\frac{\sqrt{19}}{8} \right)}\; \middle|\; n \in \mathbb{Z}\right\} \approx \left\{i \left(6.2831853 n - 1.4558354\right) - 0.60722205\; \middle|\; n \in \mathbb{Z}\right\}\end{aligned}\]

\(e\) is Euler's number, a mathematical constant that is approximately \(2.71828\).

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

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