Solve the exponential equation $ 3^{2x+1} = 81 $ and interpret the solution in the context of a population growth model where the population triples every year. - 2.13 Exponential and Logarithmic Equations and Inequalities - Review - Use this assignment in your class!

2.13 Exponential and Logarithmic Equations and Inequalities - Review

Solve the exponential equation $3^{2 x + 1} = 81$ and interpret the solution in the context of a population growth model where the population triples every year.

Solve the inequality $2^{x + 1} > 16$ and interpret the solution in the context of a financial investment that doubles every year.

Solve the exponential equation $5^{x - 2} = 25$ and interpret the solution in the context of a radioactive decay process where the substance decays to one-fifth of its original amount every year.

Solve the inequality $e^{2 x} \leq 7$ and interpret the solution in the context of a bacterial growth model where the bacteria population grows exponentially.

Solve the exponential equation $10^{3 x} = 1000$ and interpret the solution in the context of a sound intensity model where the intensity increases tenfold every unit of time.

Solve the logarithmic equation $\log_2(x+3) + \log_2(x-1) = 3$. Show all steps and verify your solution.

Solve the inequality $\log_3(x+1) > 2$. Show all steps and verify your solution.

Solve the equation $\log_5(2x - 1) = \log_5(3x - 4)$. Show all steps and verify your solution.

Solve the inequality $\log_{10}(x^2 - 4) \leq 1$. Show all steps and verify your solution.

10.

Solve the equation $\log_4(x) + \log_4(x-3) = 1$. Show all steps and verify your solution.

11.

A biologist is studying a population of bacteria that doubles every 3 hours. If the initial population is 500 bacteria, write an equation to model the population growth and determine the population after 12 hours.

12.

A car depreciates in value by 15% each year. If the initial value of the car is $20,000, write an equation to model the depreciation and determine the value of the car after 5 years.

13.

A certain radioactive substance decays at a rate of 6% per year. If the initial amount of the substance is 100 grams, write an equation to model the decay and determine the amount remaining after 10 years.

14.

A savings account offers an annual interest rate of 4%, compounded quarterly. If an initial deposit of $5000 is made, write an equation to model the account balance and determine the balance after 3 years.

15.

A certain medication decreases in the bloodstream at a rate of 20% per hour. If the initial concentration of the medication is 50 mg/L, write an equation to model the concentration and determine the concentration after 4 hours.

Solve the exponential equation $3^{2 x + 1} = 81$ and interpret the solution in the context of a population growth model where the population triples every year.

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AI InstructorAUG 12, 2024, 9:39:44 AM

First, rewrite 81 as a power of 3: $81 = 3^{4}$ . Thus, the equation becomes $3^{2 x + 1} = 3^{4}$ . Since the bases are the same, set the exponents equal: $2 x + 1 = 4$ . Solving for $x$ , we get $2 x = 3$ and $x = \frac{3}{2}$ . In the context of a population growth model, this means that it takes 1.5 years for the population to grow from its initial size to 81 times its original size.

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If the initial population is 100, what will the population be after 1.5 years?

Select one of the following options:

A.
300
B.
900
C.
8100
D.
2700

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Solve the exponential equation 32x+1=81 and interpret the solution in the context of a population growth model where the population triples every year.

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If the initial population is 100, what will the population be after 1.5 years?

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Solve the exponential equation $3^{2 x + 1} = 81$ and interpret the solution in the context of a population growth model where the population triples every year.