Question

What's the next step in this logarithmic equation?

Answer 1

\[\log \sqrt{x} = \sqrt{\log x}\]

Answer 2

\[\frac{ 1 }{ 2 } \log x = (\log x)^{\frac{ 1 }{ 2 }}\]

Answer 3

\[\frac{ 1 }{ 4 } (\log x^{2}) = \log x\]

Answer 4

\[\frac{ 1 }{ 4 } (\log x^{2}) - \log x = 0\]

Here's where I don't know what to do anymore.

Answer 5

@zepdrix

Answer 6

Woops let's correct that last step:\[\frac{ 1 }{ 4 } (\log x)^2 - \log x = 0\]The square is on the log, not the x.

Answer 7

If you make a substitution like this, \(\Large\rm u=\log x\)
it might clear up what's really going on here,\[\Large\rm \frac{1}{4}u^2-u=0\]Ehh nevermind the substitution isn't necessary.
For some reason I thought we had a trinomial.
Just factor a log(x) out of each term.

Answer 8

Oh multiply by 4 also, that 1/4 is annoying.\[\Large\rm \log x(\log x-4)=0\]

Answer 9

woah what? so if I factor it, I get \[(\log x)(\frac{ 1 }{4 } \log x - 1) = 0\] correct?

Answer 10

Ya :)

Answer 11

Then the multiplication is only applicable to the actual numbers...so \[(\log x)(\log x - 4) = 0\] thus log x = 0 or log x = 4 and so x = 1 and x = 10,000

Answer 12

Is that right?

Answer 13

Mmm yes good job! :)

Answer 14

Maybe check our answers just to make sure,\[\large\rm \log \sqrt{x} = \sqrt{\log x}\]\[\large\rm \log \sqrt{1} = \sqrt{\log 1}\qquad\implies\qquad 0=0\qquad \color{lightgreen}{\checkmark}\]
\[\large\rm \log \sqrt{10000} = \sqrt{\log 10000}\]\[\large\rm \log 100 = \sqrt{\log 10000}\]\[\Large\rm 2=\sqrt{4}\qquad\color{lightgreen}{\checkmark}\]

Answer 15

:D I see now. Thanks! I was just confused because my textbook didn't explain everything for me. xP

Answer 16

cool c: