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Comprehensive Solutions for Exponential and Logarithmic Equations in Real-Life Scenarios

This content provides a detailed exploration of solving logarithmic and exponential equations, including examples such as the natural logarithm equation ( ln(3x - 7) = 0.5 ) and the exponential equation ( 3xe^{7x} + 2 = 15 ). It consolidates methods for solving equations involving ( e^x ) and ( ln(x) ) with application to real-life situations, such as modeling rabbit populations and analyzing radioactive decay. Additionally, learn to derive models, calculate constants, and graphically represent equations.

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Comprehensive Solutions for Exponential and Logarithmic Equations in Real-Life Scenarios

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  1. Jan 10 9 i) Find the exact solutions to the equations (a) ln (3x – 7) = 5, (3) (b) 3x e7x + 2 = 15. (5)

  2. Solving ex and ln x Consolidate the methods of solving ex and ln x Extend to apply to real life situations

  3. June 09 Rabbits were introduced onto an island. The number of rabbits, P, t years after they were introduced is modelled by the equation P = 80et/5 , t ℝ, t 0.  (a) Write down the number of rabbits that were introduced to the island. (1) (b) Find the number of years it would take for the number of rabbits to first exceed 1000. (2)

  4. Jan 08 The radioactive decay of a substance is given by  R = 1000e−ct, t  0. where R is the number of atoms at time t years and c is a positive constant.  (a) Find the number of atoms when the substance started to decay. (1) It takes 5730 years for half of the substance to decay. (b) Find the value of c to 3 significant figures.(4) (c) Calculate the number of atoms that will be left when t = 22 920 (2) (d) Sketch the graph of R against t . (2)

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