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$$\log({ \log }_{ 10 }10000)$$

Steps I took to solve this:

${ \log }_{ 10 }10000=4$

${ \log }_{ 10 }4=y$

$10^{ y }=4$

${ \log }10^{ y }=\log 4$

$y=\frac { \log 4 }{ \log 10 } $

doesn't seem to come out to the correct answer...

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  • 1
    $\begingroup$ Note that $\log 10 = 1$. $\endgroup$
    – E W H Lee
    Sep 28, 2014 at 2:47
  • $\begingroup$ yes, so when I evaluate it one step further, it goes back to equaling $log4$ and $log4$ is not the correct answer $\endgroup$ Sep 28, 2014 at 2:49
  • $\begingroup$ It is correct; you can test it with your calculator. $\endgroup$
    – E W H Lee
    Sep 28, 2014 at 2:50

3 Answers 3

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Your answer is correct since $$\log (\log (10000)) = \log (\log (10^4)) = \log(4\log(10)) = \log 4$$

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  • $\begingroup$ why does the answer key state $4$ is the answer then? $\endgroup$ Sep 28, 2014 at 2:50
  • $\begingroup$ Either a mistake in the book or you misunderstood the question? $\endgroup$
    – E W H Lee
    Sep 28, 2014 at 2:52
  • $\begingroup$ The book must be wrong. $\endgroup$ Sep 28, 2014 at 2:58
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$y=\log_{10}4=\frac{\log 4}{\log 10}=\log 4$

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$\log({ \log }_{ 10 }10000)$

$=\log_{10} ( \log_{10} 10^4)$

$=\log_{10} (4 \log_{10} 10)$

$=\log_{10}4$

Yours is correct.

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