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Chapter 3: Problem 62

Find a number \(m\) such that \(\log _{5}\left(\log _{6} m\right)=3\).

Short Answer

Expert verified
The number \(m\) that satisfies the equation \(\log _{5}\left(\log _{6} m\right)=3\) is \(m = 6^{125}\).

Step by step solution

01

Rewrite the equation in exponential form

To rewrite \(\log _{5}\left(\log _{6} m\right)=3\) in exponential form, use the property \(a^{\log_{a}b} = b\), where \(a\) is the base, and \(b\) is the result. Applying this to the given equation, we get: \(5^3 = \log_{6}m\) Now, calculate \(5^3\): \(5^3 = 125\) So, the equation becomes: \(125 = \log_{6}m\)
02

Rewrite the equation in exponential form again

Now, we need to rewrite the equation \(125 = \log_{6}m\) in exponential form as well. Using the property \(a^{\log_{a}b} = b\) again, this time with a base of 6, we get: \(6^{125} = m\)
03

Simplify and find the value of m

The equation is now: \(m = 6^{125}\) Since the base and exponent are both positive integers, we can simplify this expression. The value of \(m\) is: \(m = 6^{125}\) We have now found the number \(m\) such that \(\log_{5}(\log_{6}m) = 3\).

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