Exponentials and Logarithms

Question 1

How can logₐb be interpreted?

Answer 1

logₐb can be interpreted as:

What is the power you need to raise a to to get b?

Question 2

What does logₐa equal?

Answer 2

logₐa = 1

Question 3

What does logₐ1 equal?

Answer 3

logₐ1 = 0

Question 4

What is the natural log?

Answer 4

ln() which means log base e

Question 5

Inverse functions of logs

Answer 5

aˡᵒᵍₐˣ = x
logₐ(aˣ) = x

Question 6

What are the three rules of logs?

Answer 6

logₐx + logₐy = logₐ(xy)
logₐx - logₐy = logₐ(x/y)
logₐxʸ = ylogₐx

Question 7

The exponential growth of a population of shovel-snouted lizard can be modelled by the equation 1000e^0.1t where t is the time in years.
Work out the initial population.

Answer 7

1000e^0.1t where t = 0
1000e^(0.1*0) = 1000e^0 = 1000
The initial population of shovel-snouted lizard is 1000.

Question 8

The exponential growth of a population of shovel-snouted lizard can be modelled by the equation 1000e^0.1t where t is the time in years.
How many years will it be for the population to exceed 2000 shovel-snouted lizards?

Answer 8

1000e^0.1t > 2000
e^0.1t > 2
ln(e^0.1t) > ln(2)
0.1t ln(e) > ln(2)
0.1t > ln(2)
t > ln(2) / 0.1
t > 6.9314718...
The population will have exceeded 2000 after 7 years.

Question 9

The exponential growth of a population of shovel-snouted lizard can be modelled by the equation 1000e^0.1t where t is the time in years.
Predict the population of shovel-snouted lizard after 77 years.

Answer 9

1000e^0.1t where t = 77
1000e^(0.1*77) = 1000e^7.7 = 2208347.992
Using the model the population of shovel-snouted lizard after 77 years is predicted to be 2208347.

Question 10

The exponential growth of a population of shovel-snouted lizard can be modelled by the equation 1000e^0.1t where t is the time in years.
Why might this model not be appropriate for the long term?

Answer 10

After 77 years, when t = 77, the population of shovel-snouted lizard is predicted to be above 2 million. This number is too large to be realistic - the model hasn’t taken other factors such as predators into account.