Chapter 8-Logarithm

Course Information

  • Course Code: MATH0100

  • Course Title: Pre-Calculus

  • Class Times:

    • TUES 10-12 (CP)

    • WED 10-12 (SLT2)

    • THUR 9-10 (SLT3)

  • Lecturer: Roxanne Francis

  • Office Hours: Wednesday 12-1

  • Email: roxanne.francis@uwimona.edu.jm

Assessment Structure

  • Mid-Semester Tests: 30%

  • Final Exam: 70%

Chapter Eight: Logarithm

Objectives

By the end of this chapter, students should be able to:

  1. Express sums and differences of logarithms as a single logarithm.

  2. Express a single logarithm as a sum, difference, and power of logarithms.

  3. Change bases of logarithms.

  4. Solve logarithmic equations.

  5. Apply logarithms to real-world problems.

Definition of Logarithm



  • The logarithm of a number (A) is the power (x) to which the base (b) is raised to produce the number:[ \log_b A = x \text{ if and only if } b^x = A]

Remarks on Logarithms

  • Given (\log_b A = x), where (A > 0):

    • The base (b) cannot be negative, zero, or one.

    • The common logarithm assumes base 10: (\log x = \log_{10} x).

The Natural Logarithm

  • The number (e) (approximately 2.718) is known as the natural exponent.



  • The logarithm with base (e) is the natural logarithm, represented as (\ln A):[ \log_e x = \ln x]

Relationship Between Exponential and Logarithmic Functions

  • Logarithmic functions are the inverses of exponential functions.

Examples

Evaluating Logarithms

  1. (\log_2 8 = 3) which is the same as (2^3 = 8)

  2. (\log_3 9 = 2) which is the same as (3^2 = 9)

Rules Governing Logarithms

For (M > 0) and (N > 0):

  • Difference: (\log_b M - \log_b N = \log_b \left(\frac{M}{N}\right))

  • Sum: (\log_b MN = \log_b M + \log_b N)

  • Power: (\log_b M^n = n \log_b M)

Rewriting Logarithmic Expressions

  1. Rewrite as a single logarithm:

    • (\log 5 + \log 4 = \log(5 \cdot 4))

    • (\log 12 - \log 2 = \log\left(\frac{12}{2}\right))

  2. Rewrite as a sum or difference:

    • (\log(x^2 - 4))

Changing Bases



  • To change from base (a) to base (b):[ \log_b M = \frac{\log_a M}{\log_a b}]

The Natural Logarithm in Detail



  • Natural logarithm is denoted as (\ln x) and uses the laws of logarithms.[ \log_e e = 1]

Solving Logarithmic Equations


  1. If (\log A = \log B), then (A = B).Examples include:

  • Find (x) in:

    • (\log_2 x = 4)

    • (\log_3(x + 18) + \log_3(x - 6) = 2 \log_3 x)

Real-World Applications

Example Problem






  • Growth of a patty cost over time:[ C(t) = 50e^{0.09t}]a) Find the cost in 2007.b) Predict the cost in 2009.c) Determine when the cost will be $100.

Practice Problems

  • Solve:

    1. (-2 \cdot 7^x + 5 \cdot 7^2 - 2 = 0)

    2. (6(x - 2) \cdot 6^x + 6^2 = 0)