Question
(c) Evaluate the following expression for A without using tables or \( A=\frac{4 \log 9-2 \log 27+\log 243}{7 \log 3} \)
Ask by Lang Campbell. in the United Kingdom
Mar 09,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( A = 1 \)
Solution
We start with the expression
\[
A = \frac{4 \log 9 - 2 \log 27 + \log 243}{7 \log 3}.
\]
Express each logarithm in terms of \(\log 3\) using the power rule of logarithms.
1. Since \(9 = 3^2\),
\[
4 \log 9 = 4 \log (3^2) = 4 \cdot 2 \log 3 = 8 \log 3.
\]
2. Since \(27 = 3^3\),
\[
-2 \log 27 = -2 \log (3^3) = -2 \cdot 3 \log 3 = -6 \log 3.
\]
3. Since \(243 = 3^5\),
\[
\log 243 = \log (3^5) = 5 \log 3.
\]
Now, substitute these back into the expression for \(A\):
\[
A = \frac{8 \log 3 - 6 \log 3 + 5 \log 3}{7 \log 3}.
\]
Combine the terms in the numerator:
\[
8 \log 3 - 6 \log 3 + 5 \log 3 = (8 - 6 + 5) \log 3 = 7 \log 3.
\]
Thus, we have
\[
A = \frac{7 \log 3}{7 \log 3} = 1.
\]
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Simplify this solution Bonus Knowledge
To evaluate the expression for \( A \), we can start by simplifying the logarithms. First, we know that: - \( \log 9 = \log(3^2) = 2\log 3 \) - \( \log 27 = \log(3^3) = 3\log 3 \) - \( \log 243 = \log(3^5) = 5\log 3 \) Now substitute these into the expression for \( A \): \[ A = \frac{4(2\log 3) - 2(3\log 3) + 5\log 3}{7 \log 3} \] This simplifies to: \[ A = \frac{8\log 3 - 6\log 3 + 5\log 3}{7 \log 3} \] Combining the terms in the numerator gives: \[ A = \frac{(8 - 6 + 5)\log 3}{7 \log 3} = \frac{7 \log 3}{7 \log 3} \] Since the \( \log 3 \) terms cancel out, we find: \[ A = 1 \] So, the final answer is \( A = 1 \).
