# Not understanding what is going on in this problem (evaluating a logarithm)

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

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...

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

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

• why does the answer key state $4$ is the answer then? Sep 28, 2014 at 2:50
• Either a mistake in the book or you misunderstood the question? Sep 28, 2014 at 2:52
• The book must be wrong. Sep 28, 2014 at 2:58

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

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

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

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

$=\log_{10}4$

Yours is correct.