How to solve L(x)=100 1- (0.9)×
Answer (1)
Answer:1. Finding L(x) for a Given Value of xThis is the simplest case. You're given a value for 'x' and asked to find the corresponding value of L(x).Example: Find L(5)Solution:Substitute x = 5 into the function:L(5) = 100[1 - (0.9)^5]Calculate (0.9)^5: (0.9)^5 ≈ 0.59049Substitute back into the equation:L(5) = 100[1 - 0.59049]Simplify:L(5) = 100[0.40951]L(5) ≈ 40.951Answer: L(5) ≈ 40.9512. Solving for x When Given a Value of L(x)This is more complex and requires using logarithms. You're given a value for L(x) and asked to find the corresponding value of 'x'.Example: Find the value of x when L(x) = 75Solution:Substitute L(x) = 75 into the function:75 = 100[1 - (0.9)^x]Divide both sides by 100:0.75 = 1 - (0.9)^xIsolate the exponential term:(0.9)^x = 1 - 0.75(0.9)^x = 0.25Take the logarithm of both sides. You can use either the common logarithm (log base 10) or the natural logarithm (ln):log((0.9)^x) = log(0.25) (using base 10)orln((0.9)^x) = ln(0.25) (using natural log)Use the logarithm power rule (log(a^b) = b*log(a)):x * log(0.9) = log(0.25)orx * ln(0.9) = ln(0.25)Solve for x:x = log(0.25) / log(0.9)orx = ln(0.25) / ln(0.9)Calculate:x ≈ -0.60206 / -0.04576 ≈ 13.159 (using base 10)orx ≈ -1.38629 / -0.10536 ≈ 13.159 (using natural log)Answer: x ≈ 13.159Step-by-step explanation:
