Solve for x 4^x-4^(-x)=2

Problem

4−4(−x)=2

Solution

1. Substitute a new variable to simplify the equation by letting u=4

2. Rewrite the equation in terms of u noting that 4(−x)=1/u

u−1/u=2

3. Multiply the entire equation by u to eliminate the fraction.

u2−1=2*u

4. Rearrange the equation into a standard quadratic form.

u2−2*u−1=0

5. Apply the quadratic formula u=(−b±√(,b2−4*a*c))/(2*a) where a=1 b=−2 and c=−1

u=(2±√(,(−2)2−4*(1)*(−1)))/(2*(1))

6. Simplify the expression for u

u=(2±√(,8))/2

u=(2±2√(,2))/2

u=1±√(,2)

7. Discard the negative solution because u=4 must be positive.

u=1+√(,2)

8. Substitute back 4 for u and solve for x using logarithms.

4=1+√(,2)

ln(4)=ln(1+√(,2))

x*ln(4)=ln(1+√(,2))

9. Isolate x by dividing by ln(4)

x=ln(1+√(,2))/ln(4)

Final Answer

x=ln(1+√(,2))/ln(4)

Want more problems? Check here!