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's how to solve the equation: 10. Solve the equation, (0.3)^n+1 = (0.2)^n to two decimal places. Step 1: Take the logarithm of both sides of the equation. We'll use the common logarithm (_10). _10((0.3)^n+1) = _10((0.2)^n) Step 2: Apply the logarithm power rule, _b(x^p) = p _b x, to both sides. (n+1)_10(0.3) = n_10(0.2) Step 3: Distribute _10(0.3) on the left side. n_10(0.3) + _10(0.3) = n_10(0.2) Step 4: Gather all terms containing n on one side and constant terms on the other side. n_10(0.3) - n_10(0.2) = -_10(0.3) Step 5: Factor out n from the terms on the left side. n(_10(0.3) - _10(0.2)) = -_10(0.3) Step 6: Solve for n by dividing both sides by (_10(0.3) - _10(0.2)). n = -_10(0.3)_10(0.3) - _10(0.2) Step 7: Calculate the numerical values of the logarithms and then n. Using a calculator: _10(0.3) ≈ -0.5228787 _10(0.2) ≈ -0.6989700 Substitute these values into the equation for n: n = (-(-0.5228787))/((-0.5228787) - (-0.6989700)) n = (0.5228787)/(0.6989700 - 0.5228787) n = (0.5228787)/(0.1760913) n ≈ 2.96939 Step 8: Round the value of n to two decimal places. n ≈ 2.97 The solution is n = 2.97. Last free one today — make it count tomorrow, or type /upgrade for unlimited.