Use natural logarithms to solve the exponential equations 2^x+1 =3 5^z-3=137 9e^

Question

Use natural logarithms to solve the exponential equations 2^x+1 =3 5^z-3=137 9e^x=107 2x+1=3^x Exponential Growth, use A=A_0e^kt As part of a lab experiment, Gwen needs to grow a culture of 200,000 bacteria. Which are known to double in number in 12 hrs. if she begins with 1000 bacteria,(1) find the growth rate k, and (2) (Bonus) find how many hours it takes for the culture to produce the 200,00 bacteria. After the wolf population was decimated due to overhunting, the rabbit population of a reserve began to double every 6 months. If there was an estimated 120 rabbits to begin, (1) find the growth rate k and (Bonus) (2) find the number of months required for the population to reach 2500. Exponential decay, use A=A_0e^kt The radioactive element iodine-131 has a half-life of 8 days and is often used to help diagnose patients with thyroid problems. if a certain thyroid procedure requires 0.5g and is scheduled to take place in 3 days, find the decay rate k, and (2) (Bonus) What is the minimum amount that must be on hand now (to the nearest hundredth of a gram)? Carbon-14 is a radioactive compound that occurs naturally in all living organisms, with the amount in the organism constantly renewed. After death no new carbon-14 is acquired and the amount in the organism begins to decay exponentially. If the half-life of carbon-14 is 5730 years.(1) find the decay rate k and (2)(Bonus) how old is a mummy having only 30% of the normal amount of carbon-14?

Explanation / Answer

1. a) 2^(x+1) =3

Taking natural log of both sides:

(x+1)ln2 =ln3

xln2 = ln3 -ln2

x= (ln3 -ln2)/(ln2) = 0.58

b) 5^(x-3) = 137

(x-3)ln5 = ln137

xln5 = ln137 +3ln5

x= (ln137 +3ln5)/ln5 = 6.056

c) 9e^x =107

ln(9e^x) =ln107

ln9 +x = ln107

x = ln107 -ln9 = 2.475

d) 2^(x+1) =3^x

ln(2^(x+1)) = ln(3^x)

(x+1)ln2 = xln3

x(ln2 -ln3) = -ln2

x = ln2/(ln3 -ln2) =/0.405 = 1.711

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