Solve the logarithmic equation z=ln((e^(2t))/(e^(-2t)))-arcsin((e^(-2t))/(e^(2t)))

 Exercise

$z=ln\left(\frac{e^{2t}}{e^{-2t}}\right)-arcsen\:\left(\frac{e^{-2t}}{e^{2t}}\right)$

 

Simplify the fraction $\frac{e^{2t}}{e^{-2t}}$ by $e$

$z=\ln\left(e^{\left(2t+2t\right)}\right)-\arcsin\left(\frac{e^{-2t}}{e^{2t}}\right)$

 

Simplify the fraction $\frac{e^{-2t}}{e^{2t}}$ by $e$

$z=\ln\left(e^{\left(2t+2t\right)}\right)-\arcsin\left(e^{\left(-2t-2t\right)}\right)$

 

Combining like terms $2t$ and $2t$

$z=\ln\left(e^{4t}\right)-\arcsin\left(e^{\left(-2t-2t\right)}\right)$

 

Combining like terms $-2t$ and $-2t$

$z=\ln\left(e^{4t}\right)-\arcsin\left(e^{-4t}\right)$

 

Apply the formula: $\ln\left(e^x\right)$$=x$, where $x=4t$

$z=4t-\arcsin\left(e^{-4t}\right)$

$z=4t-\arcsin\left(e^{-4t}\right)$

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