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Chapter 3: Problem 53

Suppose \(\log _{8}\left(\log _{7} m\right)=5 .\) How many digits does \(m\) have?

Short Answer

Expert verified
The integer \(m\) has 27694 digits.

Step by step solution

01

Transform the equation to exponential form using base 8

We are given the equation \(\log _{8}\left(\log _{7} m\right)=5\). The first step is to rewrite this equation as an exponential form using the base \(8\): \[8^5 = \log _{7} m\]
02

Find the value of m

Now, we need to rewrite this equation back into logarithmic form using the base \(7\) to determine the value of \(m\). Before that, let's calculate the value of \(8^5\): \[8^5 = 32768\] So the equation becomes: \[\log _{7} m = 32768\] Now we can rewrite this into the exponential form with base \(7\): \[m = 7^{32768}\]
03

Determine the number of digits in m

To find the number of digits in \(m = 7^{32768}\), we can use logarithm properties. Let's find the closest integer \(n\) such that \(10^n\) is less than or equal to \(m\), i.e., \(10^n \leq 7^{32768}\). Take logarithm base 10 on both sides: \[\log_{10} (10^n) \leq \log_{10} (7^{32768})\] Using logarithm property \(\log_a (b^c) = c\log_a b\), we get: \[n \leq 32768 \log_{10}7\] Now, we can approximate the value of \(\log_{10}7 \approx 0.845\), so we get: \[n \leq 32768 * 0.845\] \[n \leq 27694.75\] Thus, the number of digits in \(m\) is the largest integer not greater than \(27694.75\), which means that \(m\) has 27694 digits.

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