4 Given that log 20.571, log 4≈ 1.141, and log 6≈ 1.476, find log n 4 log nr (Type an integer or a decimal.)
4 Given that log 20.571, log 4≈ 1.141, and log 6≈ 1.476, find log n 4 log nr (Type an integer or a decimal.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![### Logarithmic Calculation Exercise
**Problem Statement:**
Given the following logarithmic values:
- \(\log_n 2 \approx 0.571\)
- \(\log_n 4 \approx 1.141\)
- \(\log_n 6 \approx 1.476\)
Find \(\log_n \left(\frac{4}{n}\right)\).
**Answer:**
\(\log_n \left(\frac{4}{n}\right) \approx\) [Type an integer or a decimal.]
**Explanation:**
To find the value of \(\log_n \left(\frac{4}{n}\right)\), we can use the properties of logarithms, particularly the quotient rule, which states:
\[
\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y
\]
Applying this rule, we get:
\[
\log_n \left(\frac{4}{n}\right) = \log_n 4 - \log_n n
\]
Since \(\log_n n = 1\) for any base \( n \):
\[
\log_n \left(\frac{4}{n}\right) = \log_n 4 - 1
\]
Next, substitute the given value of \(\log_n 4\):
\[
\log_n 4 \approx 1.141
\]
Therefore:
\[
\log_n \left(\frac{4}{n}\right) \approx 1.141 - 1 = 0.141
\]
So, the value is:
\[
\log_n \left(\frac{4}{n}\right) \approx 0.141
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Transcribed Image Text:### Logarithmic Calculation Exercise
**Problem Statement:**
Given the following logarithmic values:
- \(\log_n 2 \approx 0.571\)
- \(\log_n 4 \approx 1.141\)
- \(\log_n 6 \approx 1.476\)
Find \(\log_n \left(\frac{4}{n}\right)\).
**Answer:**
\(\log_n \left(\frac{4}{n}\right) \approx\) [Type an integer or a decimal.]
**Explanation:**
To find the value of \(\log_n \left(\frac{4}{n}\right)\), we can use the properties of logarithms, particularly the quotient rule, which states:
\[
\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y
\]
Applying this rule, we get:
\[
\log_n \left(\frac{4}{n}\right) = \log_n 4 - \log_n n
\]
Since \(\log_n n = 1\) for any base \( n \):
\[
\log_n \left(\frac{4}{n}\right) = \log_n 4 - 1
\]
Next, substitute the given value of \(\log_n 4\):
\[
\log_n 4 \approx 1.141
\]
Therefore:
\[
\log_n \left(\frac{4}{n}\right) \approx 1.141 - 1 = 0.141
\]
So, the value is:
\[
\log_n \left(\frac{4}{n}\right) \approx 0.141
\]
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