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
Alright ~Excel — let's do this. a) (i) Step 1: Convert the given logarithmic equations to exponential form. Given _a x = p, we can write this as x = a^p. Given _a y = q, we can write this as y = a^q. Step 2: Multiply x and y. xy = a^p · a^q Using the exponent rule a^m · a^n = a^m+n: xy = a^p+q This shows the first part. Step 3: Deduce the logarithm property. Take the logarithm with base a on both sides of the equation xy = a^p+q: _a (xy) = _a (a^p+q) Using the logarithm property _b (b^M) = M: _a (xy) = p+q Substitute back the original definitions of p and q: _a (xy) = _a x + _a y This deduces the required property. a) (ii) Step 1: Write the expression to be simplified. _4 2 + _4 32 - _4 0.25 Step 2: Use the logarithm properties _b M + _b N = _b (MN) and _b M - _b N = _b ((M)/(N)). Combine the first two terms: _4 (2 × 32) - _4 0.25 _4 64 - _4 0.25 Combine the remaining terms: _4 ((64)/(0.25)) Step 3: Simplify the argument of the logarithm. 0.25 = (1)/(4) So, _4 ((64)/(1/4)) _4 (64 × 4) _4 256 Step 4: Evaluate the logarithm. We need to find the power to which 4 must be raised to get 256. 4^1 = 4 4^2 = 16 4^3 = 64 4^4 = 256 Therefore, _4 256 = 4. The simplified value is 4. What's next?