Logarithmic function

Question

Compute the value of $f\bigl(\frac{1}{400}\bigr)$ if the function is defined as follows :

$f(xy) = f(x) + f(y)$ and $f(4)= 16$

score 1 · Answer 1 · answered Nov 01 '13 at 20:27

To answer this question, we need to assume $f$ is continuous. If $f$ is not continuous, there is no way, with the information given, to get an answer.

The relation $$ f(xy)=f(x)+f(y)\tag{1} $$ implies by induction $$ nf(x)=f(x^n)\quad\text{for }n\in\mathbb{Z}\tag{2} $$ We can infer from $(2)$ that $$ f(a^q)=qf(a)\tag{3} $$ for any $a\gt0$ and $q\in\mathbb{Q}$, and if $f$ is continuous, then $(3)$ holds for all $q\in\mathbb{R}$.

By the definition of $\log$, $(3)$ says $$ \frac{f(x)}{f(a)}=\log_a(x)\tag{4} $$

$$ \begin{align} \frac{f\left(\frac1{400}\right)}{f(4)} &=\log_4\left(\frac1{400}\right)\\ &=-\log_4(400)\\ &=-\left(2+\log_4(25)\right)\\ &=-\left(2+\log_2(5)\right)\\ f\left(\frac1{400}\right) &=-16\left(2+\log_2(5)\right)\\ &\doteq-69.1508495181978 \end{align} $$

dfeuer · Answer 2 · 2013-11-01T05:18:18.780

0

What happens when $z=1$? Does that teach you something about the behavior of $f$?

Suppose $f$ is indeed a logarithm. Then for some $b$, $$f(x)=\log_b x = \frac{\ln x}{\ln b}.$$ A little algebraic manipulation will then go a fairly long way ($b$ will go away).

edited Nov 01 '13 at 05:18

answered Nov 01 '13 at 04:55

dfeuer

9,069

1

Sadly this is the generalized function given by my research team.
Info given by them is : It is a general function with representation of f(xy) = f(x) + f(y) [I assumed it to be a logarithmic function because of its representation]. They gave a value of f(4) = 16.

So I tried it this way log 4 (base a) = 16 , from this we can get value of a as 2^1/8. Then I planning to compute log(1/400) base (2^1/8).. approximate value of f(1/400) is around -16 to -18 by looking at the graph i have

Could you help me in solving this
– user104633 Nov 01 '13 at 05:02
@user104633, my point is that this is that the third variable appears to be irrelevant. Functions of this form are all either logarithmic or extremely nasty. I'm not really understanding what you're looking for. Are you assuming it's a logarithm and just unsure how to manipulate logarithms? – dfeuer Nov 01 '13 at 05:09
@user: you might consider editing your question to include the above comment. That kind of thought/effort is highly sought after! – The Chaz 2.0 Nov 01 '13 at 05:10
I have edited the question this is one given by my research team. rest all of thing were added by me. if u can provide me with generalized approach it will be good, as i have to write a program for around 50000 values – user104633 Nov 01 '13 at 05:15

score 0 · Answer 3 · answered Nov 01 '13 at 07:18

The value 2^(1/8) you obtained for the base is correct. But for f(1/400), I am afraid that you did not look properly at the graph or the graph is wrong; the approximate value is -69.15 (its exact value on the basis of natural logarithms is - 8 Log[400] / Log[2]). Please have a look to the formulas related to the change of base for logarithms
(http://www.proofwiki.org/wiki/Change_of_Base_of_Logarithm).

Logarithmic function

3 Answers3