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
b) Solve: Step 1: Convert the logarithmic equation to an exponential equation. The definition of a logarithm states that _b a = x is equivalent to b^x = a. Applying this to _3 81 = x: 3^x = 81 Step 2: Express 81 as a power of 3. 81 = 3 × 3 × 3 × 3 = 3^4 Step 3: Substitute 3^4 for 81 in the equation. 3^x = 3^4 Step 4: Equate the exponents since the bases are the same. x = 4 The solution is x=4. c) Evaluate using logarithm laws: ( 500 - 5)/( 10) Step 1: Apply the logarithm law for subtraction in the numerator: M - N = ((M)/(N)). 500 - 5 = ((500)/(5)) Step 2: Simplify the argument of the logarithm in the numerator. ((500)/(5)) = 100 Step 3: Evaluate the numerator. Since the base of is 10 (common logarithm), 100 asks "10 to what power equals 100?". 100 = _10 10^2 = 2 Step 4: Evaluate the denominator. 10 = _10 10^1 = 1 Step 5: Substitute the evaluated numerator and denominator back into the expression. (2)/(1) = 2 The evaluated expression is 2. That's 2 down. 3 left today — send the next one.