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
Here are the solutions to your problems: 1. Solve for x: The given equation is: 4^x+1 × 2^2x-3 = 512 Step 1: Express all terms with the same base, which is 2. Since 4 = 2^2 and 512 = 2^9, substitute these into the equation: (2^2)^x+1 × 2^2x-3 = 2^9 Step 2: Apply the exponent rule (a^m)^n = a^mn. 2^2(x+1) × 2^2x-3 = 2^9 2^2x+2 × 2^2x-3 = 2^9 Step 3: Apply the exponent rule a^m × a^n = a^m+n. 2^(2x+2) + (2x-3) = 2^9 2^4x-1 = 2^9 Step 4: Since the bases are equal, equate the exponents. 4x-1 = 9 Step 5: Solve for x. 4x = 9 + 1 4x = 10 x = (10)/(4) x = (5)/(2) The value of x is 2.5. 2. Use logarithms to evaluate: Let Y = (2.735 × 0.0846)/(0.693). Step 1: Take the logarithm (base 10) of both sides. Y = ( (2.735 × 0.0846)/(0.693) ) Step 2: Apply the logarithm properties (AB) = A + B and ((A)/(C)) = A - C. Y = (2.735) + (0.0846) - (0.693) Step 3: Find the logarithm of each number using a calculator or tables. (2.735) ≈ 0.4376 (0.0846) ≈ -1.0726 (or 2.9274) (0.693) ≈ -0.1593 (or 1.8407) Step 4: Substitute these values into the equation for Y. Y = 0.4376 + (-1.0726) - (-0.1593) Y = 0.4376 - 1.0726 + 0.1593 Y = -0.6350 + 0.1593 Y = -0.4757 Step 5: Find the antilogarithm of -0.4757. Y = 10^-0.4757 Y ≈ 0.3344 The evaluated value is 0.3344.