- Expand logarithmic expressions by systematically applying the product, quotient, and power rules.
- For (a), distribute the log over the numerator and subtract the log of the denominator to get \(\boxed{8 \log(x) - 2 \log(y) + 8 \log(z)}\).
- For (b), subtract the log of each factor in the denominator to get \(\boxed{2 \log(x) - 8 \log(y) - 8 \log(z)}\).
- For (c), pull out the exponent first, then expand the interior to get \(\boxed{48 \log(x) - 48 \log(y) - 48 \log(z)}\).
Explanation
- Understanding the Problem and Logarithm Rules
We are given three logarithmic expressions to expand:
A\(\log \left(\frac{x^8 z^8}{y^2}\right)\)
B\(\log \left(\frac{x^2}{y^8 z^8}\right)\)
C\(\log \left(\left(\frac{x^8}{y^8 z^8}\right)^6\right)\)
To expand these expressions, we will use the fundamental properties of logarithms:
- Product rule: \(\log(MN) = \log(M) + \log(N)\)
- Quotient rule: \(\log\left(\frac{M}{N}\right) = \log(M) - \log(N)\)
- Power rule: \(\log(M^k) = k \log(M)\)
These rules allow us to break down complex logarithmic arguments into simpler terms involving individual variables.
- Expansion of Expression (a)
To expand \(\log \left(\frac{x^8 z^8}{y^2}\right)\), we first apply the quotient rule:
\(\log \left(\frac{x^8 z^8}{y^2}\right) = \log(x^8 z^8) - \log(y^2)\)
Next, we apply the product rule to the first term \(\log(x^8 z^8)\):
\(\log(x^8 z^8) - \log(y^2) = \log(x^8) + \log(z^8) - \log(y^2)\)
Finally, we apply the power rule to each term to bring the exponents in front of the logarithms:
\(\log(x^8) + \log(z^8) - \log(y^2) = 8 \log(x) + 8 \log(z) - 2 \log(y)\)
Thus, \(\log \left(\frac{x^8 z^8}{y^2}\right) = 8 \log(x) + 8 \log(z) - 2 \log(y)\).
- Expansion of Expression (b)
To expand \(\log \left(\frac{x^2}{y^8 z^8}\right)\), we first apply the quotient rule:
\(\log \left(\frac{x^2}{y^8 z^8}\right) = \log(x^2) - \log(y^8 z^8)\)
Then, we apply the product rule to the term \(\log(y^8 z^8)\). Note the use of parentheses as the subtraction applies to the entire product:
\(\log(x^2) - (\log(y^8) + \log(z^8)) = \log(x^2) - \log(y^8) - \log(z^8)\)
Finally, we apply the power rule:
\(\log(x^2) - \log(y^8) - \log(z^8) = 2 \log(x) - 8 \log(y) - 8 \log(z)\)
Thus, \(\log \left(\frac{x^2}{y^8 z^8}\right) = 2 \log(x) - 8 \log(y) - 8 \log(z)\).
- Expansion of Expression (c)
To expand \(\log \left(\left(\frac{x^8}{y^8 z^8}\right)^6\right)\), we first use the power rule to bring the exponent 6 outside:
\(\log \left(\left(\frac{x^8}{y^8 z^8}\right)^6\right) = 6 \log\left(\frac{x^8}{y^8 z^8}\right)\)
Next, we expand the logarithmic term inside the parentheses using the quotient and product rules:
\(6 [\log(x^8) - \log(y^8 z^8)] = 6 [\log(x^8) - (\log(y^8) + \log(z^8))]\)
\(= 6 [\log(x^8) - \log(y^8) - \log(z^8)]\)
Finally, we apply the power rule inside the brackets and then multiply by 6:
\(6 [8 \log(x) - 8 \log(y) - 8 \log(z)] = 48 \log(x) - 48 \log(y) - 48 \log(z)\)
Thus, \(\log \left(\left(\frac{x^8}{y^8 z^8}\right)^6\right) = 48 \log(x) - 48 \log(y) - 48 \log(z)\).
- Summary of Results
The expanded forms are:
A\(8 \log(x) - 2 \log(y) + 8 \log(z)\)
B\(2 \log(x) - 8 \log(y) - 8 \log(z)\)
C\(48 \log(x) - 48 \log(y) - 48 \log(z)\)
Examples
Logarithmic expansion is highly useful in data science and signal processing. For instance, when dealing with complex mathematical models involving exponential growth (like population data or sound decibels), scientists often transform multiplicative relationships into additive ones using log properties. This makes it easier to compute, linearize the model, and visualize data trends across many orders of magnitude. By expanding complex expressions into linear combinations of simpler logs, we can isolate variables and perform statistical analysis more efficiently.