Exponentials and Logs (2)

Question 1

Use logarithms to solve log_2x = 5?

Answer 1

2^(log_2x)  = 2^5
x = 32

Question 2

Use logarithms to solve 7^x = 55

Answer 2

x ln 7 = ln 55
x = ln 55 / ln 7
x = 2.059

Question 3

Reducing y=ax^n to linear form?

Answer 3

y=ax^n
log_10y = log_10a + log_10x^n
log_10y = log_10a + n log_10x
log_10y = n log_10x + log_10a
Y = mX + c

Question 4

Reducing y=kb^x to linear form

Answer 4

y=kb^x
log_10y = log_10k + log_10b^x
log_10y = log_10k + x log_10b
log_10y = (log_10b)x + log_10k 
Y = mX + c

Question 5

Where y=1.04^x

Find out how many years it takes to double the total value of an initial investment

Answer 5

2 = 1.04^x
x = ln2/ln1.04

Question 6

The number of rabbits, P, t years after they were introduced is modelled by the equation P=80e^((1/3)t)

Find the number of years it will take for the number of rabbits to exceed 1000

Answer 6

1000 = 80e^((1/3)t)
12.5 = e^((1/3)t)
ln12.5 = (1/3)t
3ln12.5 = t
t = 8 years

Question 7

The amount of a certain drug in the bloodstream t hours after it has been taken is given by the formula x = De^(-1/8)(t), where x is the amount of drug in the bloodstream in milligrams and D is the dose given in milligrams. A dose of 10mg is given and a second is given after 5 after hours. Show that the second is 13.549

Answer 7

13.549 = 10e^((-1/8)(1)) + 10e^((-1/8)(6))

Question 8

Factorise e^(-1/8)(T) + e^(-1/8)(T) * (e^(-5/8))

Answer 8

[e^(-1/8)(T)] (1+e^(-5/8))