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Chapter 3: Problem 61

Find a number \(m\) such that \(\log _{7}\left(\log _{8} m\right)=2\).

Short Answer

Expert verified
The number \(m\) that satisfies the given equation is \(m = 8^{49}\).

Step by step solution

01

Identify the given equation

We are given the equation: \(\log_{7}\left(\log_{8}m\right) = 2\)
02

Understand the relationship between outer logarithm and exponent

Recall that \(\log_{a}(b) = c\) means that \(a^c = b\). In this case, \(\log_{7}\left(\log_{8}m\right) = 2\) means that \(7^2 = \log_{8}m\)
03

Calculate the exponent

Calculate the exponent: \(7^2 = 49\). Thus, our equation now becomes: \(\log_{8}m = 49\)
04

Understand the relationship between inner logarithm and exponent

Now, recall that \(\log_{a}(b) = c\) means that \(a^c = b\). In this case, \(\log_{8}(m) = 49\) means that \(8^{49} = m\)
05

Calculate the value of m

Calculate the value of m: \(m = 8^{49}\) So, the number m that satisfies the given equation is \(m = 8^{49}\).

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