kristenjohnson1998
kristenjohnson1998 3d ago โ€ข 0 views

Logarithmic Expansion Examples with Detailed Solutions

Hey there, Math whiz! ๐Ÿ‘‹ Let's conquer logarithmic expansion together! It might seem tricky at first, but with a few examples and some practice, you'll be expanding logs like a pro in no time. Ready to jump in? ๐Ÿš€
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Uncle_Lumbago Jan 6, 2026

๐Ÿ“š Quick Study Guide

  • ๐Ÿ”‘ Product Rule: $\log_b(MN) = \log_b(M) + \log_b(N)$
  • โž— Quotient Rule: $\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)$
  • โšก Power Rule: $\log_b(M^k) = k \cdot \log_b(M)$
  • ๐Ÿ’ก Change of Base Rule: $\log_b(M) = \frac{\log_c(M)}{\log_c(b)}$
  • โž• Logarithms expand when a single logarithm contains multiple terms through multiplication, division, or exponents.

๐Ÿ“ Practice Quiz

  1. What is the expansion of $\log_2(8x)$?
    1. $\log_2(8) + \log_2(x)$
    2. $3\log_2(x)$
    3. $\log_2(8) - \log_2(x)$
    4. $\frac{\log_2(8)}{\log_2(x)}$
  2. Expand the following logarithm: $\log(100a^2b)$
    1. $2 + 2\log(a) + \log(b)$
    2. $\log(100) + \log(a) + \log(b)$
    3. $2 + \log(a^2) + \log(b)$
    4. $2 + 2\log(10a) + \log(b)$
  3. Which expression is equivalent to $\ln(\frac{x^3}{y^2})$?
    1. $3\ln(x) - 2\ln(y)$
    2. $\frac{3}{2} \ln(\frac{x}{y})$
    3. $3\ln(x) + 2\ln(y)$
    4. $\ln(3x) - \ln(2y)$
  4. Expand: $\log_5(\sqrt[3]{25x})$
    1. $\frac{2}{3} + \frac{1}{3}\log_5(x)$
    2. $\frac{1}{3}\log_5(25) + \log_5(x)$
    3. $\frac{2}{3} \log_5(x)$
    4. $\frac{1}{3}(\log_5(25) + \log_5(x))$
  5. What is the expanded form of $\log_b(\frac{x^4}{\sqrt{y}})$?
    1. $4\log_b(x) - \frac{1}{2}\log_b(y)$
    2. $\frac{4\log_b(x)}{\frac{1}{2}\log_b(y)}$
    3. $4\log_b(x) - 2\log_b(y)$
    4. $4\log_b(x) + \frac{1}{2}\log_b(y)$
  6. Simplify: $\log_3(9a^5b^{-2})$
    1. $2 + 5\log_3(a) - 2\log_3(b)$
    2. $\log_3(9) + 5\log_3(a) + 2\log_3(b)$
    3. $2 + 5\log_3(a) + 2\log_3(b)$
    4. $9 + 5\log_3(a) - 2\log_3(b)$
  7. Expand the logarithm: $\ln(\frac{e^2\sqrt{x}}{y^3})$
    1. $2 + \frac{1}{2}\ln(x) - 3\ln(y)$
    2. $2 + 2\ln(x) - 3\ln(y)$
    3. $e^2 + \frac{1}{2}\ln(x) - 3\ln(y)$
    4. $2 + \sqrt{x} - 3\ln(y)$
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