- Apply the quotient rule \(\log(M/N) = \log M - \log N\) to split the expressions.
- Use the product rule \(\log(MN) = \log M + \log N\) to expand the products within the terms.
- Distribute negative signs carefully when expanding the denominators.
- Result (a): \(\boxed{\log 9 + \log x - \log y}\)
- Result (b): \(\boxed{\log x - \log 9 - \log y}\)
Explanation
- Understanding Logarithmic Properties
To expand the logarithmic expressions, we will use the following fundamental properties of logarithms:
- Product Rule: \(\log(MN) = \log M + \log N\)
- Quotient Rule: \(\log\left(\frac{M}{N}\right) = \log M - \log N\)
These properties allow us to break down complex expressions into simpler components involving single variables or constants. We assume that \(x, y > 0\) for the expressions to be defined.
- Expanding Expression (a)
For the first expression \(\log \left(\frac{9x}{y}\right)\), we first apply the quotient rule:
\[\log \left(\frac{9x}{y}\right) = \log(9x) - \log(y)\]
Next, we apply the product rule to the term \(\log(9x)\):
\[\log(9x) = \log(9) + \log(x)\]
Substituting this back into our expression, we get:
\[\log \left(\frac{9x}{y}\right) = \log(9) + \log(x) - \log(y)\]
This is the fully expanded form.
- Expanding Expression (b)
For the second expression \(\log \left(\frac{x}{9y}\right)\), we start by applying the quotient rule:
\[\log \left(\frac{x}{9y}\right) = \log(x) - \log(9y)\]
Next, we apply the product rule to the term \(\log(9y)\), ensuring we distribute the negative sign:
\[\log(9y) = \log(9) + \log(y)\]
Substituting this back into the expression:
\[\log(x) - (\log(9) + \log(y)) = \log(x) - \log(9) - \log(y)\]
This is the fully expanded form.
- Conclusion
The expanded forms of the given logarithmic expressions are:
A\(\log \left(\frac{9x}{y}\right) = \log(9) + \log(x) - \log(y)\)
B\(\log \left(\frac{x}{9y}\right) = \log(x) - \log(9) - \log(y)\)
The final results are \(\boxed{\log(9) + \log(x) - \log(y)}\) and \(\boxed{\log(x) - \log(9) - \log(y)}\).
Examples
Logarithmic expansion is highly useful in fields like chemistry, where the pH scale is logarithmic, or in information theory, where bit rates and signal decibels are measured using logarithms. By expanding expressions, researchers can simplify complex equations involving multiplicative relationships into manageable sums and differences, allowing for easier computation and analysis of experimental data. For instance, if you were measuring the concentration of a chemical solution and needed to scale its intensity, you would use these rules to separate the factors for individual analysis.