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
**15. Without using tables, find the value of x in the equation. x^3 + 5x = 5 2 - (2/5)** Step 1: Simplify the left-hand side (LHS) of the equation using logarithm properties. The property used is a + b = (ab). x^3 + 5x = (x^3 · 5x) = (5x^3+1) = (5x^4) Step 2: Simplify the right-hand side (RHS) of the equation using logarithm properties. The properties used are n a = a^n and a - b = (a/b). 5 2 - (2/5) First, apply n a = a^n: = 2^5 - (2/5) = 32 - (2/5) Next, apply a - b = (a/b): = ((32)/(2/5)) To divide by a fraction, multiply by its reciprocal: = (32 × (5)/(2)) = (16 × 5) = 80 Step 3: Equate the simplified LHS and RHS and solve for x. (5x^4) = 80 Since the logarithms are equal and have the same base (assumed to be 10), their arguments must be equal: 5x^4 = 80 Divide both sides by 5: x^4 = (80)/(5) x^4 = 16 Take the fourth root of both sides. Since x is part of a logarithm argument ( x^3 and 5x), x must be positive. x = [4]16 x = 2 The final answer is 2.