1
$\begingroup$

I have a question regarding logarithm topic, this has bothered me for days and I could not figure things out. Here's the problem:

If $ \log_{2n}\ 2016 \ = \log_n 504\sqrt{2} $, then $n^6$ must be equal to...

I have figured several way out but still dont find the whole picture of it, it is pointless and it does not satisfy.

  • 2
    Is it log to base of n?? Please format properly. – 2017-01-25
  • 0
    yes it is, i guess there is nothing wrong with the equation. maybe its just different placement style for the base – 2017-01-25

3 Answers 3

1

We have $$\log_{2n} 2016 =\log_{n} 504\sqrt {2} $$ $$\Rightarrow \frac{\log 2016}{\log 504\sqrt {2}} =\frac {\log 2n}{\log n} $$ $$\Rightarrow \frac {\log 2016}{\log 2016-1.5\log 2} =\frac {\log 2+\log n}{\log n} $$ $$\Rightarrow \log n (\log 2016) = \log 2 (\log 2016) -1.5 (\log 2)^2 +\log n (\log 2016) -1.5 (\log 2)\log n $$ Hope you can take it from here.

1

We set them both as x. So we have the following by definition of log : $$(2n)^x=2016$$ $$n^x=504\sqrt2$$ Dividing those two equations you have $$2^x=4/\sqrt2 => 2^x=2\sqrt2 => \sqrt2 ^{2x} = \sqrt2 ^3 => x=3/2 $$ so $$n^x=504\sqrt2 => n^{3/2}=504\sqrt2 => n^6=504^4 4$$

0

Hint: write it as follows, then solve the linear equation in $\ln n$:

$$ \begin{align} \log_{2n} 2016 = \log_n 504\sqrt{2} & \quad \iff \quad \frac{\ln2016}{\ln 2n} = \frac{\ln504\sqrt{2}}{\ln n} \\ & \quad \iff \quad \frac{\ln2016}{\ln 2 + \ln n} = \frac{\ln 2016 - \frac{3}{2}\,\ln 2 }{\ln n} \end{align} $$

  • 0
    this absolutely requires calculator isn't? – 2017-01-25
  • 0
    No, it doesn't. Just work out the expression for $n$, then you'll see that all radicals vanish once you raise to the $6^{th}$ power. – 2017-01-25
  • 0
    the last expression i got is ln2 = (2/3) ln2016 – 2017-01-25
  • 0
    @IqbalJurist You lost a $\ln n$ somewhere along the way, the expression should be $\ln n = \frac{2}{3} \,\ln 2016 - \ln 2\,$ which gives $n = \sqrt[3]{2016^2} \big/ 2$. Now raise that $n$ to the $6^{th}$ power and simplify. – 2017-01-25
  • 0
    Wish the downvoter had left a comment why. – 2017-01-25