Solving Exponential Equations and Logarithms

The video tutorial explains how to solve an exponential equation to determine when a bacterial population will reach 5,000. It begins by introducing the growth equation and setting up the problem. The tutorial then demonstrates solving the equation using logarithms, specifically focusing on natural logarithms and their properties. Finally, it calculates the time required for the population to reach the desired size, concluding with an approximate time of 13.4 hours.

What is the initial population of bacteria according to the given equation?

• 10,000

• 5,000

• 1,000

• 500

What is the goal of the problem regarding the bacteria population?

• To find the initial population

• To determine the growth rate

• To estimate when the population will reach 5,000

• To calculate the population at time zero

What is the first step in solving the equation 5,000 = 1,000 x e^(0.12t)?

• Divide both sides by 1,000

• Add 1,000 to both sides

• Multiply both sides by 1,000

• Subtract 1,000 from both sides

After dividing both sides by 1,000, what does the equation become?

• 0.5 = e^(0.12t)

• 5 = e^(0.12t)

• 50 = e^(0.12t)

• 500 = e^(0.12t)

Why do we take the natural logarithm of both sides of the equation?

• To change the base of the equation

• To find the base of the logarithm

• To simplify the equation

• To eliminate the exponential

What property of logarithms allows us to move the exponent to the front?

• Product property

• Change of base property

• Quotient property

• Power property

What is the value of the natural log of e?

• e

• 1

• 0

• Undefined

What is the final step to solve for time t in the equation?

• Multiply both sides by 0.12

• Divide both sides by 0.12

• Add 0.12 to both sides

• Subtract 0.12 from both sides

What is the approximate time in hours when the population reaches 5,000?

• 14.4 hours

• 13.4 hours

• 12.4 hours

• 10.4 hours

What mathematical concept is primarily used to solve the exponential equation in this problem?

• Calculus

• Trigonometric functions

• Logarithms

• Algebraic equations