This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: State the given equation and domain restrictions. The given equation is _4(x-1) = _2(x-3). For the logarithms to be defined, their arguments must be positive: x-1 > 0 x > 1 x-3 > 0 x > 3 Combining these conditions, the valid domain for x is x > 3. Step 2: Change the base of the logarithm. Use the change of base formula _b a = (_c a)/(_c b) to convert _4(x-1) to base 2: _4(x-1) = (_2(x-1))/(_2 4) = (_2(x-1))/(2) Step 3: Substitute and simplify the equation. Substitute this back into the original equation: (_2(x-1))/(2) = _2(x-3) Multiply both sides by 2: _2(x-1) = 2 _2(x-3) Use the logarithm property n _b a = _b a^n: _2(x-1) = _2(x-3)^2 Since the bases are the same, the arguments must be equal: x-1 = (x-3)^2 Step 4: Solve the resulting quadratic equation. Expand the right side: x-1 = x^2 - 6x + 9 Rearrange into a standard quadratic form ax^2+bx+c=0: x^2 - 6x - x + 9 + 1 = 0 x^2 - 7x + 10 = 0 Factor the quadratic equation: (x-2)(x-5) = 0 This gives two potential solutions: x-2 = 0 x = 2 x-5 = 0 x = 5 Step 5: Check solutions against the domain. Recall that the valid domain for x is x > 3. For x=2: This solution is not valid because 2 3. For x=5: This solution is valid because 5 > 3. Step 6: Count the number of valid solutions. There is only one valid solution, x=5. The number of solutions is one. The final answer is One.