a Solve log2x log2x 3 2 b Show your work in detail US Dep

a) Solve log2x + log2(x - 3) = 2

b) Show your work in detail. US Department of Wildlife tagged 100 black tailed rabbits in a national forest in 2009. In 2010 they tagged 120 black tailed rabbits in the same range. If the rabbit population follows the uninhibited exponential growth model, how many rabbits will be in the range 5 years from 2009?

Solution

a. log₂x + log₂(x-3)=2

We have to use the following log rules

log a + log b= log ab

if log_ab=x then b=a^x

log₂(x(x-3))=2

   x(x-3)=2²=4

   x²-3x -4=0

   (x-4)(x+1)=0

    x=4,-1

   log cant take negative values therefore the final answer is x=4

b. We have to use the following exponential equation

       y=y₀(1+r)^t

y₀ is the population in 2009 which is 100

     120=100(1+r)¹

      120-100=100r

     20/100=r

     r= .2

Therefore rate of growth=20%=.2

So after 5 years from 2009 the population will be

    A= 100(1 +.2)⁵= 100* 1.2⁵= 248.8