Logarithm Addition

Algebra Level 3

Let $K$ be equal to the following expression. What is $2014^K?$ $K=\sum_{n=2}^{2014}\dfrac{1}{\log_n2014}$

2014!

2014^{2014}

1

2014

1 solution

Anish Puthuraya
Mar 6, 2014

Note that,
$\frac{1}{\log_{n} 2014} = \log_{2014}n$

Thus,
$K = \sum_{n=2}^{2014}\log_{2014}n$

$K = \log_{2014}2 + \log_{2014}3+\ldots+\log_{2014}2014$

$K = \log_{2014}(2\times 3\times 4\times\ldots\times 2014)$

$K = \log_{2014}2014!$

Therefore,
$2014^K = 2014^{\log_{2014}2014!} = \boxed{2014!}$

thanks for an elegant solution "anish puthuraya"

Rishabh Jain - 7 years, 3 months ago

Exactly the solution I was looking for!

Trevor B. - 7 years, 3 months ago

Are there any other methods to solve the problem? I would love to check them out.

Anish Puthuraya - 7 years, 3 months ago

Yours is the one I had in mind. I don't think there is another way to do it. The trick is to justify that $\frac{1}{\log_ab}=\log_ba$ for $a,b\neq1$ . For completion's sake, can you figure out how to do this?

Trevor B. - 7 years, 3 months ago

@Trevor B. – Well, it is pretty straightforward.

Let $\displaystyle\log_ab = k$ (say)

Then,
$b = a^k$

$\Rightarrow b^{\frac{1}{k}} = a$

Taking $\displaystyle\log$ to the base $\displaystyle b$ on both sides,

$\Rightarrow \frac{1}{k} = \log_ba$

Re-substituting the value of $\displaystyle k$ , we get,

$\frac{1}{\log_ab} = \log_ba$

Hence, Proved.

Anish Puthuraya - 7 years, 3 months ago

@Anish Puthuraya – Very good! Personally, I would say this.

$\dfrac{1}{\log_ab}\times\dfrac{\log_ba}{\log_ba}=\dfrac{\log_ba}{\log_ab\times\log_ba}=\dfrac{\log_ba}{1}=\log_ba$

You can use change of base to prove that $\log_ab\times\log_ba=1,$ as I discussed in a note I created a couple weeks ago.

Trevor B. - 7 years, 3 months ago

@Trevor B. – Yeah, that's pretty good too.

Anish Puthuraya - 7 years, 3 months ago

correct

Maya Patil - 7 years, 2 months ago

How did the final answer come from the previous step? Please...

Ashley Shamidha - 7 years ago

since a^log a b = b Therefore 2014^log 2014! equals 2014!

Huey Chan - 7 years ago

what does this ! mean?

Rahul Shah - 6 years, 11 months ago

I had the same solution but I voted up this not to watse time

Govind Balaji - 6 years, 9 months ago

Expression for K is to be found . Not of 2014^ K

prasad mahadik - 5 years, 8 months ago

Logarithm Addition

1 solution

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