Logs, exponential

Question

<sub>1.The rodent population in a city is currently estimated at 30,000 and is growing according to the Malthusian model. If it is expected to double every 7 years, when will the population reach one million? (Round your answer to one decimal place.)

2.Solve the logarithmic equation. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.)
ln 18 − ln(x − 3) = ln x

3.Solve the exponential equation using like bases
e^(x^2 + 7x) = 1/e^12</sub>

Answer 1

1) Let P(t) be the rodent population:

 

P(t) = (30,000)*(2^t/7)

 

Where t is years.  You want P(t) = 1,000,000:

 

1,000,000 = (30,000)*(2^t/7)

 33.3 = 2^t/7

 log₂(33.3) = t/7

 

7*log₂(33.3) = t ≅ 35.4 years

 

2) ln(18) - ln(x-3) = ln(x)

 

Use the identity ln(a) - ln(b) = ln(a/b)

 

ln(18/(x-3)) = ln(x)

 

Since the two logs are equal, their arguments must be equal:

 

18/(x-3) = x

18 = x(x-3)

0 = x² - 3x - 18

0 = (x-6)(x+3)

 

x = 6 and -3

 

3) e^{x^2+7x} = e^-12

 

The exponents must be equal:

 

x² + 7x = -12

x² + 7x + 12 = 0

(x+4)(x+3) = 0

 

x = -4 and -3