- The first statement, \(4 \log_3 c + 4 \log_3 d = \log_3(cd)^4\), is \(\boxed{\text{True}}\) as it directly follows from the product and power rules of logarithms.
- The second statement, \(\frac{3}{4}(\ln a + \ln b) = \ln \sqrt[3]{a^4 b^4}\), is \(\boxed{\text{False}}\) because \(\frac{3}{4}
eq \frac{4}{3}\).
- The third statement, \(3 \ln e + 2 \ln f = \ln \left(\frac{e^3}{f^2}\right)\), is \(\boxed{\text{False}}\) as it uses the product rule instead of the quotient rule.
Explanation
- Analysis and Properties of Logarithms
We need to analyze three logarithmic equations to determine if they are true or false. We will use standard properties of logarithms:
- Product Rule: \(\log_b x + \log_b y = \log_b(xy)\)
- Quotient Rule: \(\log_b x - \log_b y = \log_b(\frac{x}{y})\)
- Power Rule: \(n \log_b x = \log_b(x^n)\)
These properties hold for any positive bases and arguments.
- Evaluating Statement 1
We examine the first statement: \(4 \log_3 c + 4 \log_3 d = \log_3(cd)^4\).
Using the distributive property, the left-hand side (LHS) is:
\[4 \log_3 c + 4 \log_3 d = 4(\log_3 c + \log_3 d)\]
Applying the product rule, \(\log_3 c + \log_3 d = \log_3(cd)\), so the expression becomes:
\[4 \log_3(cd)\]
By the power rule, \(n \log_b x = \log_b(x^n)\), this simplifies to:
\[\log_3((cd)^4) = \log_3(cd)^4\]
Since the LHS equals the right-hand side (RHS), the first statement is True.
- Evaluating Statement 2
We examine the second statement: \(\frac{3}{4}(\ln a + \ln b) = \ln \sqrt[3]{a^4 b^4}\).
The LHS can be written using the product rule:
\[\frac{3}{4}(\ln a + \ln b) = \frac{3}{4} \ln(ab)\]
Applying the power rule:
\[\frac{3}{4} \ln(ab) = \ln((ab)^{3/4}) = \ln(a^{3/4}b^{3/4})\]
Now consider the RHS:
\[\ln \sqrt[3]{a^4 b^4} = \ln((a^4 b^4)^{1/3}) = \ln((ab)^4)^{1/3} = \ln((ab)^{4/3})\]
Since \(\frac{3}{4}
eq \frac{4}{3}\), the LHS is not equal to the RHS. Therefore, the second statement is False.
- Evaluating Statement 3
We examine the third statement: \(3 \ln e + 2 \ln f = \ln \left(\frac{e^3}{f^2}\right)\).
The LHS can be simplified using the power rule:
\[3 \ln e + 2 \ln f = \ln(e^3) + \ln(f^2)\]
Applying the product rule:
\[\ln(e^3) + \ln(f^2) = \ln(e^3 f^2)\]
Now consider the RHS:
\[\ln \left(\frac{e^3}{f^2}\right) = \ln(e^3) - \ln(f^2) = 3 \ln e - 2 \ln f\]
Since \(\ln(e^3 f^2)
eq \ln(e^3 / f^2)\), the third statement is False.
- Conclusion
Based on our evaluations:
- Statement 1 is True.
- Statement 2 is False.
- Statement 3 is False.
Thus, the correct answers are:
- \(4 \log_3 c + 4 \log_3 d = \log_3(cd)^4\): \(\boxed{\text{True}}\)
- \(\frac{3}{4}(\ln a + \ln b) = \ln \sqrt[3]{a^4 b^4}\): \(\boxed{\text{False}}\)
- \(3 \ln e + 2 \ln f = \ln \left(\frac{e^3}{f^2}\right)\): \(\boxed{\text{False}}\)
Examples
Logarithmic expressions are frequently used in fields like chemistry to measure acidity (pH = -\(\log[H+]\)) or in acoustics to measure sound intensity in decibels. Understanding how to manipulate these expressions allows scientists and engineers to simplify complex formulas that involve multiplicative relationships, making them easier to calculate and interpret. For instance, converting a multiplication inside a logarithm into a sum of logarithms is a key technique for simplifying calculations involving large or small scale values.