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Uncover the art of solving exponential equations and the power of logarithmic tactics. From basic to complex, learn efficient methods to crack exponential mysteries. Dive into natural growth scenarios, population models, and practical applications. Discover a wealth of problem-solving techniques that simplify exponential equations, equations with unknown powers, and logarithmic equations. Enhance your skills with trial-and-error strategies, logarithmic tactics, and the transformative power of ln in exponential equations. Delve into practical examples, such as population growth scenarios and temperature equations, to master the art of solving exponential equations.
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What is to be learned? • How to solve exponential equations
Exponential Equation? Equation where power is unknown 5x = 125 x = 3 3x = 20 x = 2.73 (General Knowledge) (Trial and error) need a better tactic
The Log Tactic 8x = 900 neat wee tactic log 8 = log 900 x = log 900 log log x log 8 x = 3.27
A bit nastier 15 X 3x = 3000 3x = 3000 3x = 200 log 3x = log 200 x log 3 = log 200 x = log 200 x = 4.82 15 log 3
Solving Exponential Equations Equations where unknown is a power
30 X 6x = 1200 Eliminate 30 6x = 1200 6x = 40 Logs on both Sides Log 6x = log 40 Neat wee tactic x log 6 = log 40 x = log 40 30 (using log10) = 2.06 log 6
Natural Growth P is Population over t days Population of some sort of creepy crawlie P = P0e0.2t If there are 1000 ccs to start with, how long will it take to reach a million? 1000000 = 1000e0.2t 1000000 = e0.2t 1000 = e0.2t ln 1000 = ln e0.2t ln 1000 = 0.2t ln e P0 P ln 1000 = 0.2t 1000 t = ln 1000 0.2 = 34.5 days = 1
When e is involved using ln is a neat tactic Tina’s tea temperature (T 0C) after t mins T = T0e-0.2t How long until it is half its initial temp? T = e-0.2t 0.5 = e-0.2t ln 0.5 = ln e-0.2t ln 0.5 = -0.2t ln e ln 0.5 = -0.2t t = ln 0.5 T0 = 1 = 3.5 mins -0.2
