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: Combine the logarithmic terms. The given equation is _x 2x + _x 54 = 5. Using the logarithm property _b M + _b N = _b (MN), we can combine the terms on the left side: _x (2x · 54) = 5 _x (108x) = 5 Step 2: Convert the logarithmic equation to an exponential equation. The definition of a logarithm states that if _b A = C, then b^C = A. Applying this to our equation: x^5 = 108x Step 3: Solve the exponential equation for x. Rearrange the equation to set it to zero: x^5 - 108x = 0 Factor out x: x(x^4 - 108) = 0 This gives two possible cases: Case 1: x = 0 Case 2: x^4 - 108 = 0 x^4 = 108 Step 4: Check for valid values of x. For a logarithm _b A to be defined, the base b must satisfy b > 0 and b ≠ 1. Also, the argument A must satisfy A > 0. Case 1: If x=0, the base of the logarithm would be 0, which is not allowed. Also, the term 2x would be 0, which is not allowed as an argument. Therefore, x=0 is not a valid solution. Case 2: If x^4 = 108, then x = ± [4]108. Since the base x must be positive, we take the positive root: x = [4]108 Let's check if this value satisfies the conditions for the base: • x = [4]108 is clearly greater than 0. • x = [4]108 is not equal to 1 (since 1^4 = 1 and 2^4 = 16, 3^4 = 81, 4^4 = 256, [4]108 is between 3 and 4). The arguments 2x and 54 will both be positive if x is positive. Thus, x = [4]108 is the only valid solution. The final answer is x = [4]108. Last free one today — make it count tomorrow, or type /upgrade for unlimited.