Exponential Equations and Logarithms

This video tutorial covers the exponential function e, its unique properties, and its inverse, the natural logarithm ln. It explains how e is a special number in modeling exponential growth and decay, and how its value is equal to its rate of change at any point. The video also discusses the inverse relationship between e and ln, showing how they reflect over the line y = x. It provides methods for solving equations involving e and ln, with step-by-step examples and practice problems to reinforce learning.

What is the approximate value of the mathematical constant e?

• 1.61803

• 1.41421

• 2.71828

• 3.14159

Why is the number e considered special in modeling exponential growth?

• It is an integer.

• It is a prime number.

• Its value is equal to its rate of change at any point.

• It is the largest number.

Where does the graph of y = e^x cross the y-axis?

• At (1, 1)

• At (0, 1)

• At (1, 0)

• At (0, 0)

What is the inverse of the exponential function y = e^x?

• y = log x

• y = x^2

• y = 1/x

• y = ln x

What does ln x represent?

• Logarithm base 2 of x

• Logarithm base e of x

• Logarithm base 5 of x

• Logarithm base 10 of x

How is the inverse of a function represented on a graph?

• As a reflection in the line y = x

• As a translation along the x-axis

• As a rotation around the origin

• As a translation along the y-axis

What happens when you apply ln to both sides of the equation e^x = 5?

• You get x = 5

• You get x = ln 5

• You get x = e^5

• You get x = 1/5

What is the result of applying e to both sides of the equation ln x = 10?

• x = 10

• x = e^10

• x = 1/10

• x = ln 10

In the equation e^(3x + 5) = 8, what is the first step to isolate the exponential term?

• Subtract 5 from both sides

• Multiply both sides by 3

• Divide both sides by 3

• Add 5 to both sides

After isolating the exponential term in e^(3x) = 3, what is the next step?

• Divide both sides by 3

• Multiply both sides by 3

• Apply e to both sides

• Apply ln to both sides