%5E2-10-4%5Ex-8-%3D3*4%5Ex-36.jpeg "(4^x-8)^2-10 4^x-8 =3*4^x-36")
Solving the Equation: (4^x-8)^2-104^x-8 = 34^x-36
In this article, we will solve the equation (4^x-8)^2-104^x-8 = 34^x-36. This equation involves exponentiation, quadratic terms, and linear terms, making it a challenging problem to solve.
Step 1: Expand the Left-Hand Side
First, we need to expand the left-hand side of the equation using the exponent rule (a-b)^2 = a^2 - 2ab + b^2.
(4^x-8)^2 = (4^x)^2 - 2(4^x)(8) + 8^2 = 4^(2x) - 16*4^x + 64
Step 2: Simplify the Left-Hand Side
Now, we simplify the left-hand side by combining like terms.
4^(2x) - 164^x + 64 - 104^x - 8 = 4^(2x) - 26*4^x + 56
Step 3: Equate the Two Expressions
Next, we equate the simplified left-hand side with the right-hand side of the equation.
4^(2x) - 264^x + 56 = 34^x - 36
Step 4: Move All Terms to One Side
We move all terms to the left-hand side and set the equation equal to zero.
4^(2x) - 29*4^x + 92 = 0
Step 5: Factor the Quadratic Expression
Now, we try to factor the quadratic expression.
(4^x - 4)(4^x - 23) = 0
Step 6: Solve for x
Finally, we solve for x by setting each factor equal to zero and solving for x.
4^x - 4 = 0 --> 4^x = 4 --> x = 1
4^x - 23 = 0 --> 4^x = 23 --> x = log4(23)
Therefore, the solutions to the equation are x = 1 and x = log4(23).
