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Chapter 3: Problem 60

Find a number \(n\) such that \(\log _{3}\left(\log _{2} n\right)=2\).

Short Answer

Expert verified
The value of \(n\) that satisfies the given condition is \(n = 512\).

Step by step solution

01

Remove the outer logarithm

To remove the outer \(\log_{3}\), we can use the fact that the inverse of a logarithm function is an exponential function. Therefore, we will use exponentiation with base \(3\) on both sides of the equation: \[3^{\log_{3}\left(\log_{2} n\right)} = 3^2.\] Due to the fact that the logarithm and exponential functions are inverses, we are left with: \[\log_{2} n = 3^2.\]
02

Simplify the exponential

Now, we will calculate the value of \(3^2\), which equals \(9\). So, we have: \[\log_{2} n = 9.\]
03

Remove the inner logarithm

Similarly to step 1, we can remove the \(\log_2\) by using exponentiation with base \(2\) on both sides of the equation: \[2^{\log_{2}n} = 2^9.\] As before, the logarithm and exponential functions are inverses, so we are left with: \[n = 2^9.\]
04

Calculate the value of 'n'

Finally, we will calculate the value of \(2^9\), which equals \(512\). Therefore, the value of \(n\) that satisfies the given condition is: \[n = 512.\]

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