Express as a single logarithm and, if possible, simplify. 1 log b + 3 log c 2 1 log b +3 log c= 2 (Use parentheses to indicate the argument of the logarithm.)
Express as a single logarithm and, if possible, simplify. 1 log b + 3 log c 2 1 log b +3 log c= 2 (Use parentheses to indicate the argument of the logarithm.)
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Logarithmic Simplification Problem
Express as a single logarithm and, if possible, simplify.
\[
\frac{1}{2} \log b + 3 \log c
\]
---
\[
\frac{1}{2} \log b + 3 \log c = \left[ \text{Input Box} \right]
\]
(Use parentheses to indicate the argument of the logarithm.)
---
#### Explanation:
This problem requires simplifying a combination of logarithms into a single logarithm expression. The coefficients will be used as exponents according to logarithmic properties:
\[
\frac{1}{2} \log b + 3 \log c = \log(b^{\frac{1}{2}}) + \log(c^3)
\]
Using the property that \(\log A + \log B = \log (A \cdot B)\), the expression can be further simplified to:
\[
= \log \left( b^{\frac{1}{2}} \cdot c^3 \right)
= \log \left( \sqrt{b} \cdot c^3 \right)
\]
Therefore, the final simplified form is:
\[
\boxed{\log(\sqrt{b} \cdot c^3)}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa3230e3d-dd38-497e-8c1e-7c1c5570828c%2Fdb194382-e9d0-4f84-81c7-99cb741c1b36%2Fae0nfwg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Logarithmic Simplification Problem
Express as a single logarithm and, if possible, simplify.
\[
\frac{1}{2} \log b + 3 \log c
\]
---
\[
\frac{1}{2} \log b + 3 \log c = \left[ \text{Input Box} \right]
\]
(Use parentheses to indicate the argument of the logarithm.)
---
#### Explanation:
This problem requires simplifying a combination of logarithms into a single logarithm expression. The coefficients will be used as exponents according to logarithmic properties:
\[
\frac{1}{2} \log b + 3 \log c = \log(b^{\frac{1}{2}}) + \log(c^3)
\]
Using the property that \(\log A + \log B = \log (A \cdot B)\), the expression can be further simplified to:
\[
= \log \left( b^{\frac{1}{2}} \cdot c^3 \right)
= \log \left( \sqrt{b} \cdot c^3 \right)
\]
Therefore, the final simplified form is:
\[
\boxed{\log(\sqrt{b} \cdot c^3)}
\]
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