- Expand logarithmic expressions by using the quotient rule \(\log(\frac{M}{N}) = \log M - \log N\), product rule \(\log(MN) = \log M + \log N\), and power rule \(\log(M^k) = k \log M\).
- Apply these properties step-by-step for each sub-problem:
A\(\log \left(\frac{x^5 z^5}{\sqrt{y}}\right) = \boxed{5 \log x + 5 \log z - \frac{1}{2} \log y}\)
B\(\log \left(\frac{\sqrt{x}}{y^5 z^5}\right) = \boxed{\frac{1}{2} \log x - 5 \log y - 5 \log z}\)
C\(\log \left(\sqrt{\frac{x^5}{y^5 z^5}}\right) = \boxed{\frac{5}{2} \log x - \frac{5}{2} \log y - \frac{5}{2} \log z}\)
Explanation
- Logarithmic Expansion Principles
To expand the given logarithmic expressions, we will use three fundamental properties of logarithms:
- Product Rule: \(\log(MN) = \log(M) + \log(N)\)
- Quotient Rule: \(\log(\frac{M}{N}) = \log(M) - \log(N)\)
- Power Rule: \(\log(M^k) = k \log(M)\)
We also treat square roots as exponents of \(1/2\), noting that \(\log(\sqrt{M}) = \log(M^{1/2}) = \frac{1}{2} \log(M)\). Using these, we can break down complex expressions into simpler terms involving \(\log x\), \(\log y\), and \(\log z\).
- Expansion of Expression (a)
For the first expression, \(\log \left(\frac{x^5 z^5}{\sqrt{y}}\right)\):
- First, apply the quotient rule: \(\log(x^5 z^5) - \log(\sqrt{y})\).
- Next, apply the product rule to the first term and the power rule for roots to the second term: \(\log(x^5) + \log(z^5) - \log(y^{1/2})\).
- Finally, use the power rule on all components:
\(5 \log x + 5 \log z - \frac{1}{2} \log y\).
- Expansion of Expression (b)
For the second expression, \(\log \left(\frac{\sqrt{x}}{y^5 z^5}\right)\):
- Apply the quotient rule: \(\log(\sqrt{x}) - \log(y^5 z^5)\).
- Apply the product rule to the second term and express the root as a power: \(\log(x^{1/2}) - (\log(y^5) + \log(z^5))\).
- Distribute the negative sign and apply the power rule to all terms:
\(\frac{1}{2} \log x - 5 \log y - 5 \log z\).
- Expansion of Expression (c)
For the third expression, \(\log \left(\sqrt{\frac{x^5}{y^5 z^5}}\right)\):
- Start by applying the power rule for the outer root: \(\frac{1}{2} \log \left(\frac{x^5}{y^5 z^5}\right)\).
- Apply the quotient rule: \(\frac{1}{2} (\log(x^5) - \log(y^5 z^5))\).
- Apply the product rule to the term \(\log(y^5 z^5)\) and the power rule: \(\frac{1}{2} (5 \log x - (5 \log y + 5 \log z))\).
- Finally, distribute the \(1/2\) factor:
\(\frac{5}{2} \log x - \frac{5}{2} \log y - \frac{5}{2} \log z\).
Examples
Logarithmic expansion is essential in fields like acoustics or chemistry. For example, when calculating pH levels (\(pH = -\log[H^+]\)) or decibel levels, complex formulas involving multiple variables are often simplified into sums or differences. By breaking down products and ratios into individual terms, scientists can analyze how changing one variable (like the concentration of a chemical) independently affects the overall result, simplifying data interpretation and complex calculations.