Question

Solve for x.

Answer 1

Answer:

Option: C is correct.

$x=\ln\sqrt{\dfrac{t+1}{1-t}}$

Step-by-step explanation:

We are asked to solve for 'x' such that we are given a equation as:

$\dfrac{e^x-e^{-x}}{e^x+e^{-x}}=t$

This equation could also be written as:

$e^x-e^{-x}=t(e^x+e^{-x})\\\\e^x-e^{-x}=te^x+te^{-x}\\\\e^x-te^x=te^{-x}+e^{-x}\\\\(1-t)e^x=(t+1)e^{-x}\\\\e^{2x}=\dfrac{t+1}{1-t}$

on taking logarithmic function on both the sides we have:

$2x=\ln (\dfrac{t+1}{1-t})$

(since $\ln (e^x)=x$ )

$x=\dfrac{1}{2}\times \ln (\dfrac{t+1}{1-t})$  

as we know $\dfrac{1}{2}\times \ln b = \ln (b^\frac{1}{2})$

Hence, we have: $x=\ln\sqrt{\dfrac{t+1}{1-t}}$

Hence, option C is correct.

Answer 2

Answer:

OPTION C

Step-by-step explanation: