Express as sums and differences: The expression presented is a logarithmic function:10. $ \log_5 \left( \frac{\sqrt[4]{d^4} \cdot e^6 \cdot f^8}{\sqrt[4]{a^3} \cdot b^3 \cdot c^3} \right) $Explanation:- This expression involves the logarithm base $5$ of a fraction.- In the numerator, we have the fourth root of $d^4$, multiplied by $e$ raised to the sixth power, and $f$ raised to the eighth power.- In the denominator, there is the fourth root of $a^3$, multiplied by $b$ cubed, and $c$ cubed.This structure involves manipulation of powers and roots inside a logarithmic function, useful for various algebraic simplifications and calculations.

The given term is log5d4e6f84a3b3c34. First write the term as differences using the property…

Answered: 10. log 5 y dt e 6 fB 4Va3 b3 c3

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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Express as sums and differences:

The expression presented is a logarithmic function:

10. $ \log_5 \left( \frac{\sqrt[4]{d^4} \cdot e^6 \cdot f^8}{\sqrt[4]{a^3} \cdot b^3 \cdot c^3} \right) $

Explanation:
- This expression involves the logarithm base $5$ of a fraction.
- In the numerator, we have the fourth root of $d^4$, multiplied by $e$ raised to the sixth power, and $f$ raised to the eighth power.
- In the denominator, there is the fourth root of $a^3$, multiplied by $b$ cubed, and $c$ cubed.

This structure involves manipulation of powers and roots inside a logarithmic function, useful for various algebraic simplifications and calculations.

Expert Solution

Step 1

The given term is $\log_{5} (\frac{\sqrt[4]{d^{4} e^{6} f^{8}}}{\sqrt[4]{a^{3} b^{3} c^{3}}})$ .

First write the term as differences using the property $\log_{c} (\frac{a}{b}) = \log_{c} (a) - \log_{c} (b)$ .

$\begin{array}{rcl} \log_{5} (\frac{\sqrt[4]{d^{4} e^{6} f^{8}}}{\sqrt[4]{a^{3} b^{3} c^{3}}}) & = & \log_{5} (\sqrt[4]{d^{4} e^{6} f^{8}}) - \log_{5} (\sqrt[4]{a^{3} b^{3} c^{3}}) \\ = & \log_{5} ({(d^{4})}^{\frac{1}{4}} {(e^{6})}^{\frac{1}{4}} {(f^{8})}^{\frac{1}{4}}) - \log_{5} ({(a^{3})}^{\frac{1}{4}} {(b^{3})}^{\frac{1}{4}} {(c^{3})}^{\frac{1}{4}}) \\ = & \log_{5} (d e^{\frac{3}{2}} f^{2}) - \log_{5} (a^{\frac{3}{4}} b^{\frac{3}{4}} c^{\frac{3}{4}}) \end{array}$

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