Logarithms: Expanding and Condensing

Learning Outcomes

  • Expand logarithms using properties of logarithms.

  • Condense logarithmic expressions into a single logarithm.

Expanding Logarithms

  • Properties of Logs: Include the product rule, quotient rule, and power rule.

  • Combining Rules: Often multiple rules are applied to expand expressions.

  • Example:
    logb(6xy) = logb(6x) + logb(y) = logb(6) + logb(x) - logb(y)

Rules for Expanding:

  1. Product Rule: (logb(A imes C) = logb(A) + logb(C))

  2. Quotient Rule: (logb(A/C) = logb(A) - logb(C))

  3. Power Rule: (logb(A^n) = n logb(A))

Example Expansion

  • Expand (ln(x^4y^7)):

    1. Start with the quotient rule:
      ln(x^4y^7) = ln(x^4) + ln(y) - ln(7)

    2. Then, apply the power rule:
      ln(x^4) + ln(y) - ln(7) = 4ln(x) + ln(y) - ln(7)

Tips for Success

  • Practice different combinations of rules on varying expressions.

  • Pay attention to the base of the logarithms; they must be the same to condense effectively.

Condensing Logarithms

  • Combine sums and differences of logarithms with the same base into a single logarithm.

  • Important: All logarithms must have the same base.

Steps to Condense:

  1. Apply the power rule first to convert coefficients into exponents.

  2. Use the product rule for sums and the quotient rule for differences.

Example of Condensing:

  • Combine log3(5) + log3(8) - log3(2):

    • Use product rule:
      log3(5 imes 8) - log3(2) = log3(40) - log3(2)

    • Then apply the quotient rule:
      log3(40/2) = log3(20)

Complex Logarithmic Expressions

  • Use rules systematically to condense complex logarithmic expressions:
    log2(x^2) + (1/2) log2(x - 1) - 3 log2((x + 3)^2)

Example Solution:

  • Apply power rule:
    log2(x^2) + log2((x - 1)^{1/2}) - log2((x + 3)^6)

  • Now combine:
    log2(x^2 * √(x - 1) / (x + 3)^6)

Applications of Logarithms

  • pH and Logarithms: pH = -log10(aH+), where aH+ is the hydrogen ion activity.

  • Example:
    For aH+ = 2.5 imes 10^{-6},
    pH = - log_{10}(2.5 * 10^{-6}) → pH = 5.6

Impact of Concentration Change on pH:

  • If hydrogen ion concentration doubles:
    pH = P - log(2) ≈ P - 0.301

  • This implies close relationships between ion concentration and pH value changes in solutions.

Summary

  • Understanding the rules for logarithms and their properties is crucial for both expanding and condensing logarithmic expressions.