Question

Solve In (x2 + 4) – 2ln (x) = 1 for the variable x-

Answer 1

Answer::

`x=(2)/(√(e-1))\text{ or }x=-\frac{2}{\sqrt[]{e-1}}`

Step-by-step explanation:

Given the equation:

`\ln (x^2+4)-2\ln (x)=1`

First, we can make the number 2 an index:

`ln(x^2+4)-\ln (x^2)=1`

Apply the division law of logarithm to combine the left-hand side:

`\ln ((x^2+4)/(x^2))=1`

Take the exponent of both sides:

`\begin{gathered} e^{\ln ((x^2+4)/(x^2))}=e^1 \\ (x^2+4)/(x^2)=e \\ x^2+4=ex^2 \end{gathered}`

Solve the equation above for x:

`\begin{gathered} ex^2-x^2=4 \\ x^2(e-1)=4 \\ x^2=(4)/(e-1) \\ x=\pm\sqrt[]{(4)/(e-1)} \\ x=\pm\frac{2}{\sqrt[]{e-1}} \end{gathered}`

The values of the variable x are:

`x=(2)/(√(e-1))\text{ or }x=-\frac{2}{\sqrt[]{e-1}}`