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Chapter 1: Problem 87

The domain of \(\mathrm{f}(\mathrm{x})=\log _{5}\left[\log _{6}\left(\log _{8} \mathrm{x}\right)\right]\) is (a) \(x>4\) (b) \(x>8\) (c) \(x<8\) (d) \(x<4\)

Short Answer

Expert verified
The domain of the function \(\mathrm{f}(\mathrm{x})=\log _{5}\left[\log _{6}\left(\log _{8}\mathrm{x}\right)\right]\) is \(x>8\). Therefore, the correct answer is (b) \(x>8\).

Step by step solution

01

Identify the first logarithm

The first logarithm we encounter is \(\log _{8}\mathrm{x}\). In order for the function to be defined, we need \(\mathrm{x}\) to be greater than 0: \(\mathrm{x} > 0\)
02

Identify the second logarithm

The second logarithm is \(\log _{6}\left(\log _{8}\mathrm{x}\right)\). According to the first step, we know that the argument of this logarithm is already positive. In order for the second logarithm to be well-defined, we need the argument of the second logarithm to be positive too: \(\log _{8}\mathrm{x} > 0\) We solve this inequality: \(8^0 =1\) since any non-zero number raised to the power of 0 equals 1. Therefore, we get: \(\mathrm{x} > 1\)
03

Identify the third logarithm

The third logarithm is \(\log _{5}\left[\log _{6}\left(\log _{8}\mathrm{x}\right)\right]\). According to step 2, we know that the argument of this logarithm is already positive. In order for the third logarithm to be well-defined, we need the argument of the third logarithm to be positive too: \(\log _{6}\left(\log _{8}\mathrm{x}\right) > 0\) We solve this inequality: \(6^0 =1\) Therefore, we get: \(\log _{8}\mathrm{x} > 1\) Now, we solve this new inequality: \(8^1 =8\) Therefore, we get: \(\mathrm{x} > 8\)
04

Identify the domain

Now, we combine the results from step 1, step 2 and step 3: \(\mathrm{x} > 0\) \(\mathrm{x} > 1\) \(\mathrm{x} > 8\) Since we need to satisfy all three inequalities, the valid domain would be the most restrictive one, which is \(\mathrm{x} > 8\). Therefore, the domain of the function \(\mathrm{f}(\mathrm{x})=\log _{5}\left[\log _{6}\left(\log _{8}\mathrm{x}\right)\right]\) is \(x>8\). Looking at the given options, the correct answer is: (b) \(x>8\)

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Most popular questions from this chapter

Which one of the following relations on \(R\) is an equivalence relation? (a) a \(\mathrm{R}_{1} \mathrm{~b} \Leftrightarrow|\mathrm{a}|=|\mathrm{b}|\) (b) a \(\mathrm{R}_{2} \mathrm{~b} \Leftrightarrow \mathrm{a} \geq \mathrm{b}\) (c) a \(\mathrm{R}_{3} \mathrm{~b} \Leftrightarrow\) a divides \(\mathrm{b}\) (d) a \(\mathrm{R}_{4} \mathrm{~b} \Leftrightarrow \mathrm{a}<\mathrm{b}\)

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