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Let \(f(x) = \log_{3}(x)\)and \(g(x) = 3^x\).

What is the value of\(f(g(f(f(f(g(27))))))?\)

The solution of the equation \(7^{x+7} = 8^x\)can be expressed in the form \(x = \log_{b}\left(7^7\right).\)

What is \(b\)?

 

Solve for \(n\) given that \(\log_{64}\left(\dfrac 1 2\right) + \log_{64}(1) + \log_{64}(2) + \log_{64}(4) + \log_{64}(8) + \log_{64}(n) = 2.\)

 May 3, 2023
 #1
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1. f(g(f(f(f(g(27)))))) = 3.

 May 3, 2023
 #2
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2. b = 56.

 May 3, 2023
 #3
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For the first one, we can make use of the fact that f and g are inverses, so we can cancel to get f(f(27))=f(3)=1.

For the second one, set 

\(7^x\cdot7^7=8^x\)

\(7^7=\frac{8^x}{7^x}=\frac{8}{7}^x\)

\(log_\frac{8}{7}{7^7}=x\)

So b is \(\frac{8}{7}\).

For the third one, the expression is equal to

-1/6 + 0 + 1/6 + 1/3 + 1/2 + \(log_{64}{n}\)=2

\(log_{64}{n}=\frac{7}{6}\)

\(n=128\)

 May 4, 2023
 #4
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JamesJohnson  May 10, 2023
edited by JamesJohnson  May 20, 2023

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