Derivatives - Properties of ln(x)

  • The natural logarithm function, ln(x), is the inverse of the exponential function, e^x.
  • It is defined for positive values of x.
  • The derivative of ln(x) is given by: d/dx ln(x) = 1/x.
  • The properties of ln(x) include the following:
  1. Property 1: ln(1) = 0
  1. Property 2: ln(e) = 1
  1. Property 3: ln(xy) = ln(x) + ln(y)
  1. Property 4: ln(x/y) = ln(x) - ln(y)
  1. Property 5: ln(x^r) = r ln(x) Let’s look at some examples to better understand these properties. Example 1: Find the derivative of f(x) = ln(x) Solution: Using the property d/dx ln(x) = 1/x, we have: f’(x) = d/dx ln(x) = 1/x Example 2: Evaluate ln(e^3) Solution: Using property 2: ln(e) = 1, we have: ln(e^3) = 3 Example 3: Simplify ln(5x^2y) Solution: Using property 3: ln(xy) = ln(x) + ln(y), we have: ln(5x^2y) = ln(5) + ln(x^2) + ln(y) Example 4: Simplify ln(2x/y^3) Solution: Using property 4: ln(x/y) = ln(x) - ln(y), we have: ln(2x/y^3) = ln(2x) - ln(y^3) Example 5: Find the derivative of g(x) = ln(x^2) Solution: Using property 5: ln(x^r) = r ln(x), we have: g’(x) = d/dx ln(x^2) = 2 ln(x) These properties of ln(x) are useful in various applications of calculus.

Properties of ln(x)

  • ln(x) is defined only for positive values of x.
  • The natural logarithmic function, ln(x), is the inverse function of e^x.
  • The graph of ln(x) is always increasing.
  • The domain of ln(x) is (0, +∞).
  • The range of ln(x) is (-∞, +∞).

Property 1: ln(1) = 0

  • ln(1) is equal to zero.
  • This property can be easily derived from the fact that e^0 = 1.

Property 2: ln(e) = 1

  • ln(e), where e is Euler’s number, is equal to 1.
  • This property arises from the fact that e is the base of the natural logarithm.

Property 3: ln(xy) = ln(x) + ln(y)

  • ln(xy) is equal to ln(x) plus ln(y).
  • This property allows us to simplify expressions involving multiplication inside the logarithm.

Property 4: ln(x/y) = ln(x) - ln(y)

  • ln(x/y) is equal to ln(x) minus ln(y).
  • This property allows us to simplify expressions involving division inside the logarithm.

Property 5: ln(x^r) = r ln(x)

  • ln(x^r) is equal to r times ln(x).
  • This property helps simplify expressions with exponentiation inside the logarithm.

Example 1

Find the derivative of f(x) = ln(x) Solution: Using the property d/dx ln(x) = 1/x, we have: f’(x) = d/dx ln(x) = 1/x

Example 2

Evaluate ln(e^3) Solution: Using property 2: ln(e) = 1, we have: ln(e^3) = 3

Example 3

Simplify ln(5x^2y) Solution: Using property 3: ln(xy) = ln(x) + ln(y), we have: ln(5x^2y) = ln(5) + ln(x^2) + ln(y)

Example 4

Simplify ln(2x/y^3) Solution: Using property 4: ln(x/y) = ln(x) - ln(y), we have: ln(2x/y^3) = ln(2x) - ln(y^3)

Example 5

Find the derivative of g(x) = ln(x^2) Solution: Using property 5: ln(x^r) = r ln(x), we have: g’(x) = d/dx ln(x^2) = 2 ln(x)

Applications of ln(x)

  • ln(x) is commonly used in various fields, such as physics, economics, and engineering.
  • It is used to model exponential growth or decay.
  • It is used to solve differential equations.
  • It is used in calculus to find the area under a curve.

Example 6

Solve the differential equation dy/dx = k/y, where k is a constant. Solution: Rearranging the equation, we have: y dy = k dx Integrating both sides: ∫ y dy = ∫ k dx Solving the integrals: (y^2)/2 = kx + C Rearranging the equation: y^2 = 2kx + C

Example 7

Find the area under the curve y = ln(x) from x = 1 to x = 2. Solution: To find the area under the curve, we can integrate the function y = ln(x) over the given interval. Using the property ∫ ln(x) dx = x(ln(x) - 1) + C, we have: ∫ ln(x) dx = [x(ln(x) - 1)] evaluated from x = 1 to x = 2 Evaluating the definite integral: ∫ ln(x) dx = [2(ln(2) - 1)] - [1(ln(1) - 1)] Simplifying: ∫ ln(x) dx = 2(ln(2) - 1)

Summary

  • The natural logarithm function, ln(x), is the inverse of the exponential function, e^x.
  • Its derivative is 1/x.
  • The properties of ln(x) include ln(1) = 0, ln(e) = 1, ln(xy) = ln(x) + ln(y), ln(x/y) = ln(x) - ln(y), and ln(x^r) = r ln(x).
  • ln(x) has a domain of (0, +∞) and a range of (-∞, +∞).
  • It is used in various applications, such as modeling exponential growth, solving differential equations, and finding areas under curves.

Conclusion

  • Understanding the properties of ln(x) is essential in the study of calculus.
  • ln(x) has several useful properties that simplify calculations and allow us to solve various problems.
  • It is important to practice applying these properties in order to become proficient in working with ln(x) functions.
  • Further study and practice will strengthen your understanding of ln(x) and its applications in calculus.

References

  1. Stewart, J. (2015). Calculus: Early Transcendentals. Cengage Learning.
  1. Larson, R., & Edwards, B. (2013). Calculus. Cengage Learning.
  1. Anton, H., Bivens, I., & Davis, S. (2012). Calculus: Early Transcendentals. John Wiley & Sons.

Questions

  1. Find the derivative of h(x) = ln(3x^4).
  1. Evaluate ln(1/e).
  1. Simplify ln(2xy^2).
  1. Find the area under the curve y = ln(x) from x = 1 to x = e.
  1. Solve the differential equation dy/dx = 1/x.

Questions (continued)

  1. Find the derivative of f(x) = 3 ln(x^2).
  1. Evaluate ln(e^5).
  1. Simplify ln(x^2/y^3).
  1. Find the area under the curve y = ln(x) from x = e to x = 2e.
  1. Solve the differential equation dy/dx = kx, where k is a constant.

Thank You!

  • Thank you for attending this lecture on the properties of ln(x).
  • If you have any questions, please feel free to ask.
  • Don’t forget to practice the examples and review the concepts covered today.
  • Good luck with your studies!