exponential function. y = a x if we look at the graphs of y = a x for different values of a, we can...

Exponential Function

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Exponential Function

y = ax

If we look at the graphs of y = ax for different values of a, we can see that the gradient of the exponential graph increases as the value of a increases

a gradient

Page 3: Exponential Function. y = a x If we look at the graphs of y = a x for different values of a, we can see that the gradient of the exponential graph increases

y = ax

a increasing

Page 4: Exponential Function. y = a x If we look at the graphs of y = a x for different values of a, we can see that the gradient of the exponential graph increases

• Gradient of y = 2x at (0,1) is 0.693

• Gradient of y = 3x at (0,1) is 1.099

• So there must be a value (between 2 and 3) where the gradient at (0,1) is exactly 1

• This number is 2.71828… and is called e

Page 5: Exponential Function. y = a x If we look at the graphs of y = a x for different values of a, we can see that the gradient of the exponential graph increases

Natural Logarithms

Page 6: Exponential Function. y = a x If we look at the graphs of y = a x for different values of a, we can see that the gradient of the exponential graph increases

By drawing the graph of y = ex we can see that there is an inverse function

Page 7: Exponential Function. y = a x If we look at the graphs of y = a x for different values of a, we can see that the gradient of the exponential graph increases

Page 8: Exponential Function. y = a x If we look at the graphs of y = a x for different values of a, we can see that the gradient of the exponential graph increases

If y = ex

Take logs of both sides logey = logeex

logey = xlogee but logee = 1

So x = logey

Interchanging x and y gives y = logex

(written as y = lnx)

This is the inverse function

Page 9: Exponential Function. y = a x If we look at the graphs of y = a x for different values of a, we can see that the gradient of the exponential graph increases

Questions

1. Make x the subject of lnx – lnA = kt

2. Make t the subject of s = e-kt

3. Make x the subject of y-5 = (A - 5)ex

Page 10: Exponential Function. y = a x If we look at the graphs of y = a x for different values of a, we can see that the gradient of the exponential graph increases

Questions4. A colony of human settler on a previously

uninhabited planet. After t years, their population, P, is given by

P = 100e0.056t

a. Sketch the graph of P against t b. How many settlers were there planet

initially? c. How long does it take for the

population to reach 1 million?

Page 11: Exponential Function. y = a x If we look at the graphs of y = a x for different values of a, we can see that the gradient of the exponential graph increases

Answers1. lnx – lnA = kt ln(x/A) = kt

x/A = ekt

x = Aekt

2. s = e-kt lns =-kt t =-lns/k

3. y-5 = (A - 5)ex ex =(y-5)/(A - 5) x =ln((y-5)/(A - 5))

Page 12: Exponential Function. y = a x If we look at the graphs of y = a x for different values of a, we can see that the gradient of the exponential graph increases

Answers ctd4. P = 100e0.056t

b. When t= 0 P = 100e0 = 100

c. P = 100e0.056t = 1,000,000 e0.056t = 10,000

0.056t = ln 10,000= 9.2103 t = 9.2103/0.056 = 164.47

.47 years =12x.47 months = 5.64monthsIt takes 164.5 years to 4 s.f. or 164 years 6 months to the nearest month

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