Question

Solve the equation for x, accurate to three decimal places: e4x − e2x = 12.x = 1.280x = 1.866x = 0.549x = 0.693

Answer 1

Given:

`e^(4x)-e^(2x)=12`

To Determine: The value of x

Solution

`\begin{gathered} e^(4x)-e^(2x)=12 \\ (e^x)^4-(e^x)^2=12 \end{gathered}`
`\begin{gathered} let:a=e^x \\ Therefore, \\ (e^x)^4-(e^x)^2=12 \\ a^4-a^2=12 \\ a^4-a^2-12=0 \end{gathered}`

Solve the derived equation by factorizing completing

`\begin{gathered} a^4-a^2-12=0 \\ a^4-4a^2+3a^2-12=0 \\ a^2(a^2-4)+3(a^2-4)=0 \\ (a^2-4)(a^2+3)=0 \\ a^2-4=0,or,a^2+3=0 \end{gathered}`
`\begin{gathered} a^2-4=0 \\ a^2-2^2=0 \\ diiference\text{ of two square is expanded as} \\ a^2-b^2=(a-b)(a+b) \\ Therefore \\ a^2-2^2=0 \\ (a-2)(a+2)=0 \\ a-2=0,or,a+2=0 \\ a=2,a=-2 \end{gathered}`
`\begin{gathered} Also,a^2+3=0 \\ a^2=-3(No\text{ solution because square root of a negative number is an imaginary number\rparen} \end{gathered}`

Therefore

`a=2,or,a=-2`
`\begin{gathered} e^x=a \\ e^x=2,or,e^x=-2(no\text{ solution\rparen} \\ x=ln2 \\ x=0.693 \end{gathered}`

Hence, the value of x is 0.693