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# log problems

0
55
4

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
#3
+1

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
+1

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