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Properties of Logarithms quiz #1 Flashcards

Properties of Logarithms quiz #1
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  • What is the product rule for logarithms, and how is it used to expand log expressions?
    The product rule for logarithms states that log_b(mn) = log_b(m) + log_b(n). It is used to expand a logarithm of a product into the sum of two separate logarithms with the same base.
  • How does the quotient rule for logarithms help in condensing log expressions?
    The quotient rule states that log_b(m/n) = log_b(m) - log_b(n). It allows you to condense the difference of two logarithms with the same base into a single logarithm of a quotient.
  • Explain the power rule for logarithms and provide an example.
    The power rule states that log_b(m^n) = n * log_b(m). For example, log_2(x^3) = 3 * log_2(x).
  • How can you change the base of a logarithm, and why is this useful?
    You can change the base of a logarithm using the formula log_b(m) = log_a(m) / log_a(b). This is useful for evaluating logs with bases not available on calculators by converting them to common (base 10) or natural (base e) logs.
  • What is the value of log_b(b) and log_b(1), and why?
    log_b(b) = 1 because b^1 = b, and log_b(1) = 0 because b^0 = 1. These follow from the definition of logarithms as inverses of exponentials.
  • What is the product rule for logarithms and how is it applied to expand log expressions?
    The product rule states that log_b(mn) = log_b(m) + log_b(n), allowing you to expand a logarithm of a product into the sum of two separate logarithms with the same base.
  • How does the quotient rule for logarithms help in condensing log expressions?
    The quotient rule states that log_b(m/n) = log_b(m) - log_b(n), so you can condense the difference of two logs with the same base into a single log of a quotient.
  • Explain the power rule for logarithms and give an example.
    The power rule states that log_b(m^n) = n * log_b(m); for example, log_2(x^3) = 3 * log_2(x).
  • How can you change the base of a logarithm, and why is this useful?
    You can change the base using log_b(m) = log_a(m) / log_a(b), which is useful for evaluating logs with bases not available on calculators by converting them to base 10 or base e.
  • What are the values of log_b(b) and log_b(1), and why?
    log_b(b) = 1 because b^1 = b, and log_b(1) = 0 because b^0 = 1; these follow from the definition of logarithms as inverses of exponentials.