Solve for x (2^x-2^(-x))/3=4

Problem

(2^x−2^(−x))/3=4

Solution

1. Multiply both sides by 3 to isolate the numerator.

(2^x−2^(−x))/3=4

2^x−2^(−x)=12

2. Substitute u=2^x to transform the equation into a quadratic form, noting that 2^(−x)=1/u

u−1/u=12

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

u^2−1=12*u

4. Rearrange the equation into standard quadratic form a*u^2+b*u+c=0

u^2−12*u−1=0

5. Apply the quadratic formula u=(−b±√(,b^2−4*a*c))/(2*a) to solve for u

u=(12±√(,(−12)^2−4*(1)*(−1)))/(2*(1))

u=(12±√(,144+4))/2

u=(12±√(,148))/2

6. Simplify the radical and the fraction.

u=(12±2√(,37))/2

u=6±√(,37)

7. Back-substitute u=2^x and discard the negative root, since 2^x must be positive (6−√(,37)<0.

2^x=6+√(,37)

8. Solve for x by taking the logarithm of both sides.

(log_2)(2^x)=(log_2)(6+√(,37))

x=(log_2)(6+√(,37))

Final Answer

(2^x−2^(−x))/3=4⇒x=(log_2)(6+√(,37))

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