Write the logarithmic expression as a single logarithm with coefficient 1, and simplify as much as possible. Assume that the variables represent positive real numbers. 3 log, m-4log; n+8logsp Part: 0/3 Part 1 of 3 Apply the power property of logarithms. = log-log-log;
Write the logarithmic expression as a single logarithm with coefficient 1, and simplify as much as possible. Assume that the variables represent positive real numbers. 3 log, m-4log; n+8logsp Part: 0/3 Part 1 of 3 Apply the power property of logarithms. = log-log-log;
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Simplifying Logarithmic Expressions
**Problem:**
Write the logarithmic expression as a single logarithm with coefficient 1, and simplify as much as possible. Assume that the variables represent positive real numbers.
\[ 3\log_5 m - 4\log_5 n + 8\log_5 p \]
**Solution Steps:**
#### Part: 0 / 3
(no input required by the user in this part)
#### Part 1 of 3
**Step 1: Apply the power property of logarithms.**
The power property of logarithms states that \(a \log_b x = \log_b x^a\).
Starting expression:
\[ 3\log_5 m - 4\log_5 n + 8\log_5 p \]
Applying the power property:
\[ = \log_5 m^3 - \log_5 n^4 + \log_5 p^8 \]
**Diagram Explanation:**
There is a graphical representation involving multiple input boxes and a text area. Each box should be correctly filled with the following transpositions:
\[ = \log_5 \, \square - \log_5 \, \square + \log_5 \, \square \]
From the problem:
\[ = \log_5 m^3 - \log_5 n^4 + \log_5 p^8 \]
Each \(\square\) should be filled as follows:
\[ = \log_5 m^3 - \log_5 n^4 + \log_5 p^8 \]
When combining these logarithmic expressions into a single logarithm using the properties of logarithms, the expression becomes:
\[ = \log_5 \left(\frac{m^3 p^8}{n^4}\right) \]
Finally, the simplified single logarithm expression is:
\[ \log_5 \left(\frac{m^3 p^8}{n^4}\right) \]
Thus:
\[ 3\log_5 m - 4\log_5 n + 8\log_5 p = \log_5 \left(\frac{m^3 p^8}{n^4}\right) \]
This is the final simplified form of the given logarithmic expression.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc9a4113c-7e46-4999-90ad-a989e5f085c8%2F216dfd63-84e4-48d9-814d-fe02cb3f497d%2Fr31az39_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Simplifying Logarithmic Expressions
**Problem:**
Write the logarithmic expression as a single logarithm with coefficient 1, and simplify as much as possible. Assume that the variables represent positive real numbers.
\[ 3\log_5 m - 4\log_5 n + 8\log_5 p \]
**Solution Steps:**
#### Part: 0 / 3
(no input required by the user in this part)
#### Part 1 of 3
**Step 1: Apply the power property of logarithms.**
The power property of logarithms states that \(a \log_b x = \log_b x^a\).
Starting expression:
\[ 3\log_5 m - 4\log_5 n + 8\log_5 p \]
Applying the power property:
\[ = \log_5 m^3 - \log_5 n^4 + \log_5 p^8 \]
**Diagram Explanation:**
There is a graphical representation involving multiple input boxes and a text area. Each box should be correctly filled with the following transpositions:
\[ = \log_5 \, \square - \log_5 \, \square + \log_5 \, \square \]
From the problem:
\[ = \log_5 m^3 - \log_5 n^4 + \log_5 p^8 \]
Each \(\square\) should be filled as follows:
\[ = \log_5 m^3 - \log_5 n^4 + \log_5 p^8 \]
When combining these logarithmic expressions into a single logarithm using the properties of logarithms, the expression becomes:
\[ = \log_5 \left(\frac{m^3 p^8}{n^4}\right) \]
Finally, the simplified single logarithm expression is:
\[ \log_5 \left(\frac{m^3 p^8}{n^4}\right) \]
Thus:
\[ 3\log_5 m - 4\log_5 n + 8\log_5 p = \log_5 \left(\frac{m^3 p^8}{n^4}\right) \]
This is the final simplified form of the given logarithmic expression.
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