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
You're on a roll — Here's the solution for question 4: a) Evaluate _2 128. Step 1: Express 128 as a power of 2. 128 = 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2^7 Step 2: Substitute this into the logarithm expression. _2 128 = _2 (2^7) Step 3: Use the logarithm property _b (b^k) = k. _2 (2^7) = 7 The final answer is 7. b) Simplify _3 ((x^2)/(sqrt(z))). Step 1: Apply the logarithm property _b ((M)/(N)) = _b M - _b N. _3 ((x^2)/(sqrt(z))) = _3 (x^2) - _3 (sqrt(z)) Step 2: Rewrite sqrt(z) as z^1/2. = _3 (x^2) - _3 (z^1/2) Step 3: Apply the logarithm property _b (M^k) = k _b M. = 2_3 x - (1)/(2)_3 z The final answer is 2_3 x - (1)/(2)_3 z. c) Solve for x: 5^2x-1 = 125. Step 1: Express 125 as a power of 5. 125 = 5 × 5 × 5 = 5^3 Step 2: Substitute this into the equation. 5^2x-1 = 5^3 Step 3: Since the bases are equal, equate the exponents. 2x-1 = 3 Step 4: Solve the linear equation for x. 2x = 3+1 2x = 4 x = (4)/(2) x = 2 The final answer is 2. What's next?