College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter5: Exponential And Logarithmic Functions
Section5.3: Logarithmic Functions And Their Graphs
Problem 140E
Related questions
Question
![### Logarithm Problem
#### Problem Statement:
Given that \( \log_m 2 \approx 0.59 \), \( \log_m 4 \approx 1.18 \), and \( \log_m 6 \approx 1.525 \), find \( \log_m \left( \frac{4}{m} \right) \).
#### Solution:
To solve this problem, we will use the properties of logarithms.
We need to find:
\[ \log_m \left( \frac{4}{m} \right) \]
Using the logarithm property that states \( \log_b \left( \frac{x}{y} \right) = \log_b x - \log_b y \):
\[ \log_m \left( \frac{4}{m} \right) = \log_m 4 - \log_m m \]
We know from the given information:
- \( \log_m 4 \approx 1.18 \)
- \( \log_m m = 1 \) (since the logarithm of a number to its own base is always 1)
Substituting these values into the equation:
\[ \log_m \left( \frac{4}{m} \right) = 1.18 - 1 = 0.18 \]
Therefore,
\[ \log_m \left( \frac{4}{m} \right) \approx 0.18 \]
### Answer:
\[ \log_m \left( \frac{4}{m} \right) \approx 0.18 \]
Please type the answer as a decimal in the box provided: \( \boxed{0.18} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F455afe53-baf3-4322-8440-d4ba87f3dd5f%2F62ff9626-9815-4567-8ddf-d0d4fecafb6e%2Fzszok7s_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Logarithm Problem
#### Problem Statement:
Given that \( \log_m 2 \approx 0.59 \), \( \log_m 4 \approx 1.18 \), and \( \log_m 6 \approx 1.525 \), find \( \log_m \left( \frac{4}{m} \right) \).
#### Solution:
To solve this problem, we will use the properties of logarithms.
We need to find:
\[ \log_m \left( \frac{4}{m} \right) \]
Using the logarithm property that states \( \log_b \left( \frac{x}{y} \right) = \log_b x - \log_b y \):
\[ \log_m \left( \frac{4}{m} \right) = \log_m 4 - \log_m m \]
We know from the given information:
- \( \log_m 4 \approx 1.18 \)
- \( \log_m m = 1 \) (since the logarithm of a number to its own base is always 1)
Substituting these values into the equation:
\[ \log_m \left( \frac{4}{m} \right) = 1.18 - 1 = 0.18 \]
Therefore,
\[ \log_m \left( \frac{4}{m} \right) \approx 0.18 \]
### Answer:
\[ \log_m \left( \frac{4}{m} \right) \approx 0.18 \]
Please type the answer as a decimal in the box provided: \( \boxed{0.18} \).
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