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The function e x and its inverse, lnx

y=3 x. y=e x. y=2 x. x. The function e x and its inverse, lnx. The functions like y = 2 x , y = 3 x are called exponential functions because the variable x is the power (exponent or index) of a base number. The graph of y = 2 x , y = 3 x and y = e x.

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The function e x and its inverse, lnx

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  1. y=3x y=ex y=2x x The function ex and its inverse, lnx The functions like y = 2x, y = 3x are called exponential functions because the variable x is the power (exponent or index) of a base number. The graph of y = 2x, y = 3x and y = ex It is clear from the graph that the number e is somewhere between 2 and 3 but closer to 3 than 2. This is a special number and its value correct to 8 decimal places is e =2.18281828.

  2. y 1 x 1 Graph of ex and lnx y = ex y = x y = lnx y = ex y = lnx

  3. Evaluating function ex and lnx Evaluate (i) e2 (ii) e-3 (iii) ln0.5 (iv) ½ ln10 (i) 7.39 (ii) 0.0498 (iii) 7.39 (iv) 1.15 Find the value of x (i) ln ex = 3 (ii) elnx = 5 (iii) e2lnx = 16 (iv) e-lnx = ½ (i) x = 3 (ii) x = 5 (iii) x = 4 (iv) x = 2

  4. Solving equations involving ex and lnx Solve for x (i) 3e2x – 1 = 54 (ii) 3e2x –5ex = 2  3e2x = 55 Let y = ex  e2x = 18.333..  3y2 – 5y = 2  2x= ln18.333..  3y2 – 5y – 2 = 0  2x= 2.9087..  (3y )(y )  x= 1.45  (3y + 1)(y - 2)  y = - 1/3 or y = 2  ex = 2  x = 0.693

  5. Solving equations involving ex and lnx Solve for x (i) ln(3x – 5) = 3.4 (ii) ln(3x + 1) – ln3 = 1  ln((3x + 1)/3) = 1  3x – 5 = e3.4  ((3x + 1)/3) = e1  3x – 5 = 29.964…  ((3x + 1)/3) = 2.718…  3x = 34.964…  x= 11.7.  3x + 1 = 8.1548..  3x = 7.1548… x = 2.38

  6. Exponential Decay/Growth A quantity N is decreasing such that at time t (a) Find the value of N when t = 5 (b) Find the value of t when t = 2 = 18.4 (a) (b) t = 16.1

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