- Use the power rule to expand \(\log(x^y)\) as \(y \log(x)\).
- Use the product rule to expand \(\log(xy)\) as \(\log(x) + \log(y)\).
- Use the quotient rule to expand \(\log(x/y)\) as \(\log(x) - \log(y)\).
- The final answers are: a. \(\boxed{y \log(x)}\), b. \(\boxed{\log(x) + \log(y)}\), c. \(\boxed{\log(x) - \log(y)}\).
Explanation
- Introduction to Logarithmic Properties
The logarithmic properties are essential tools that allow us to simplify or expand logarithmic expressions by converting complex operations (like exponentiation, multiplication, and division) into simpler operations (like multiplication, addition, and subtraction). For these derivations, we assume \(x > 0\) and \(y > 0\), ensuring the expressions are well-defined within the domain of the logarithm. We will use the fundamental power rule, product rule, and quotient rule to expand the given expressions.
- Expanding the Power Expression
The power rule for logarithms states that for any real number \(y\), \(\log(x^y) = y \log(x)\). This property allows us to "bring down" the exponent as a multiplier in front of the logarithm. Thus, the expansion is \(\log(x^y) = y \log(x)\).
- Expanding the Product Expression
The product rule for logarithms states that the logarithm of a product is equal to the sum of the logarithms of the individual factors: \(\log(xy) = \log(x) + \log(y)\). This property is highly useful for breaking down complex products into simpler components. Thus, the expansion is \(\log(xy) = \log(x) + \log(y)\).
- Expanding the Quotient Expression
The quotient rule for logarithms states that the logarithm of a quotient is equal to the difference between the logarithm of the numerator and the logarithm of the denominator: \(\log\left(\frac{x}{y}\right) = \log(x) - \log(y)\). This allows us to separate terms involving division. Thus, the expansion is \(\log\left(\frac{x}{y}\right) = \log(x) - \log(y)\).
- Final Conclusion
Combining these results, we have successfully expanded the three expressions using the core properties of logarithms. The final expanded forms are as follows:
A\(\log(x^y) = y \log(x)\)
B\(\log(xy) = \log(x) + \log(y)\)
C\(\log\left(\frac{x}{y}\right) = \log(x) - \log(y)\)
Examples
Logarithms are used extensively in fields like science and finance to handle numbers that span many orders of magnitude. For instance, in chemistry, the pH scale (which measures how acidic or basic a substance is) is defined using logarithms, where \(\text{pH} = -\log[H^+]\). Similarly, in acoustics, sound intensity is measured in decibels (dB), which is calculated using the formula \(L_I = 10 \log_{10}(\frac{I}{I_0})\). Understanding how to expand these expressions helps professionals manipulate these formulas to simplify data processing and theoretical modeling.