# Thread: growth function and prove

1. ## growth function and prove

[IMG]file:///C:/DOCUME%7E1/ADMINI%7E1/LOCALS%7E1/Temp/moz-screenshot.png[/IMG]Can you help for the functions and prove at the attachment ?

2. Originally Posted by isiksoy7
[IMG]file:///C:/DOCUME%7E1/ADMINI%7E1/LOCALS%7E1/Temp/moz-screenshot.png[/IMG]Can you help for the functions and prove at the attachment ?

This is basic logarithms stuff, I think: use that $\log_ax=\frac{\ln x}{\ln a}$ , so

$\frac{f(n)}{g(n)}=\frac{\frac{\ln n}{\ln a}}{\frac{\ln n}{\ln b}}=\frac{\ln b}{\ln a}$

Tonio

3. The key is to show that $\log_an=C_1\log_bn$ and $\log_bn=C_2\log_an$ for some constants $C1, C2$.

Now logrithm is not such a scary beast. Personally, I remember only three properties of logarithms.

(1) Definition: $\log_an=x$ iff $a^x=n$
(2) $\log_a(x^y)=y\log_ax$
(3) $\log_a(xy)=\log_ax+\log_ay$.

It's not difficult to deduce (2) and (3) from (1), so the main thing you need to remember is that logarithm is an inverse function to power: $a^{\log_ax}=x$ and $\log_a(a^x)=x$.

So, how to express $\log_an$ through $\log_bn$? Let $\log_an=x$. By (1), $a^x=n$. We need to use $\log_bn$, so let's take $\log_b$ of both sides: $\log_b(a^x)=\log_bn$. By (2), $x\log_ba=\log_bn$. Recalling the definition of $x$, we get $\log_ba\log_an=\log_bn$.

It's also easy to remember this last formula. Here is a mnemonic (not mathematical) explanation: you go from $b$ to $a$ ( $\log_ba$), then from $a$ to $n$ ( $\log_an$) and the result is from $b$ to $n$ ( $\log_bn$).

4. Thanks Emakarov for your helps.