Combines the product, quotient, and power rules to expand single, complex logarithmic expressions into simpler terms, and conversely, to condense multiple logarithmic terms into a single expression.
What is the goal of 'expanding' a logarithmic expression?
To rewrite a single logarithm with a complex argument as a sum, difference, and/or multiple of simpler logarithms.
Fully expand the expression: \(\log_b(x^2y)\).
$$2\log_b(x) + \log_b(y)$$
Fully expand the expression: \(\ln(\frac{a^5}{b})\).
$$5\ln(a) - \ln(b)$$
Fully expand the expression: \(\log(\frac{x^3\sqrt{y}}{z^4})\).
$$3\log(x) + \frac{1}{2}\log(y) - 4\log(z)$$
Fully expand the expression: \(\log_5(\sqrt[3]{25x^2})\).
First, apply power rule to the cube root: \(\frac{1}{3}\log_5(25x^2)\). Then apply product rule: \(\frac{1}{3}(\log_5(25) + \log_5(x^2))\). Simplify and apply power rule again: \(\frac{1}{3}(2 + 2\log_5(x)) = \frac{2}{3} + \frac{2}{3}\log_5(x)\).
What is the goal of 'condensing' a logarithmic expression?
To rewrite a sum, difference, or multiple of logarithms as a single logarithm with a single argument.
Condense into a single logarithm: \(3\log_b(x) + 5\log_b(y)\).
First, apply the Power Rule: \(\log_b(x^3) + \log_b(y^5)\). Then, apply the Product Rule: \(\log_b(x^3 y^5)\).
Condense into a single logarithm: \(2\ln(a) - \frac{1}{2}\ln(b)\).
First, apply the Power Rule: \(\ln(a^2) - \ln(b^{\frac{1}{2}})\). Then, apply the Quotient Rule: \(\ln(\frac{a^2}{\sqrt{b}})\).
Condense into a single logarithm: \(\log(x) + \log(y) - \log(z)\).
Work from left to right. Product Rule first: \(\log(xy) - \log(z)\). Then Quotient Rule: \(\log(\frac{xy}{z})\).
Condense into a single logarithm: \(3\log_4(x) - (\log_4(y) + 2\log_4(z))\).
Simplify the parenthesis first: \(3\log_4(x) - (\log_4(y) + \log_4(z^2)) = 3\log_4(x) - \log_4(yz^2)\). Apply power and quotient rules: \(\log_4(x^3) - \log_4(yz^2) = \log_4(\frac{x^3}{yz^2})\).
