OR WHICH NO WORK ISy"z16 24z24log.x5

Answered: 16 24 8 log. Z. to

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter5: Orthogonality

Section5.3: The Gram-schmidt Process And The Qr Factorization

Problem 11AEXP

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Question

OR WHICH NO WORK IS
y"z
16 24
z24
log.
x5

Expert Solution

Step 1

Given

$\log_{a} (\sqrt[8]{\frac{y^{16} z^{24}}{x^{5}}})$

Step 2

To simplify $\log_{a} (\sqrt[8]{\frac{y^{16} z^{24}}{x^{5}}})$ .

Now rewrite $\log_{a} (\sqrt[8]{\frac{y^{16} z^{24}}{x^{5}}})$ as $\log_{a} ({(\frac{y^{16} z^{24}}{x^{5}})}^{\frac{1}{8}})$

Now apply the log rule $\log_{a} (x^{b}) = b \log_{a} (x)$ .

$\log_{a} ({(\frac{y^{16} z^{24}}{x^{5}})}^{\frac{1}{8}}) = \frac{1}{8} \log_{a} (\frac{y^{16} z^{24}}{x^{5}})$

Now use the rule $\log_{a} (\frac{m}{n}) = \log_{a} m - \log_{a} n$

$\log_{a} ({(\frac{y^{16} z^{24}}{x^{5}})}^{\frac{1}{8}}) = \frac{1}{8} \log_{a} (\frac{y^{16} z^{24}}{x^{5}}) = \frac{1}{8} ((\log_{a} (y^{16} z^{24})) - \log_{a} (x^{5})) = \frac{1}{8} ((8 \log_{a} (y^{2} z^{3})) - 5 \log_{a} (x)) = \frac{(8 \log_{a} (y^{2} z^{3}))}{8} - \frac{5 \log_{a} (x)}{8} = \log_{a} (y^{2} z^{3}) - \frac{5}{8} \log_{a} (x)$

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